#486693

order	512
nilpotency class	4
isoclinism	isoclinic class of #486689
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#18203
commutator, Frattini, cosocle	C₂×M₁₆	C₂⁴
prime radical	#62275	C₂
LCS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×#131 ◁ G
LECS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

674 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 674 conjugacy classes of subgroups — too many to draw. Only its 130 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 64 points:

(1,2,7,30,8,13,4,10)(3,14,15,39,18,44,31,42)(5,21,24,11,25,52,19,38)(6,26,29,12,28,40,20,41)(9,27,32,17,35,45,34,49)(16,46,22,43,48,62,54,63)(23,55,57,51,56,37,50,61)(33,47,59,53,60,64,36,58);(1,3)(2,9)(4,15)(5,22)(6,27)(7,31)(8,18)(10,32)(11,36)(12,39)(13,35)(14,40)(16,24)(17,20)(19,48)(21,33)(23,47)(25,54)(26,44)(28,45)(29,49)(30,34)(37,43)(38,59)(41,42)(46,61)(50,53)(51,62)(52,60)(55,63)(56,64)(57,58);(1,4,8,7)(2,10,13,30)(3,15,18,31)(5,19,25,24)(6,20,28,29)(9,32,35,34)(11,21,38,52)(12,40,41,26)(14,42,44,39)(16,22,48,54)(17,45,49,27)(23,50,56,57)(33,59,60,36)(37,61,55,51)(43,46,63,62)(47,53,64,58);(1,5,4,19,8,25,7,24)(2,11,30,52,13,38,10,21)(3,16,31,54,18,48,15,22)(6,23,20,50,28,56,29,57)(9,33,32,59,35,60,34,36)(12,37,40,61,41,55,26,51)(14,43,39,62,44,63,42,46)(17,47,27,58,49,64,45,53)

Polycyclic presentation:

2^9|a^2=gh,c^2=i,d^2=gi,e^2=g,g^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eg,g^a=gi,h^a=hi,d^b=dg,e^b=egi,f^b=fhi,g^b=gi,e^c=eh,f^e=fi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250033, #72250076, #72250223, …
maximal normal subgroup of	#72250033, #72250076, #72250223, …
maximal quotient of	#72250377, #72279512

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 393889);