#484982

order	512
nilpotency class	4
isoclinism	isoclinic class of #12304
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1616
commutator, Frattini, cosocle	C₂×M₁₆	C₂⁴
socle	C₂	C₂×#1616
prime radical	#60915	C₂
LCS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×M₁₆ ◁ G
LECS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

673 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 673 conjugacy classes of subgroups — too many to draw. Only its 136 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307091, #72307445, #72307452
maximal normal subgroup of	#72307091, #72307445, #72307452

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 392178);