#393938

order	512
nilpotency class	3
isoclinism	isoclinic class of #393937
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#14715
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#63090	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3855 conjugacy classes of subgroups; only the 296 normal subgroups are shown.

normal subgroups only

This group has 3855 conjugacy classes of subgroups — too many to draw. Only its 296 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:

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

Polycyclic presentation:

2^9|a^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,g^b=gi,h^b=hi,e^c=ei,g^c=gi,h^c=hi,e^d=ei,f^d=fi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72286176, #72286194, #72286231
maximal normal subgroup of	#72286176, #72286194, #72286231

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 301134);