#504164

order	512
nilpotency class	4
isoclinism	isoclinic class of #499769
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#135	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1772 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

513 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 72
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 72 points:

(1,2,5,7)(3,6,4,8)(9,10,15,22)(11,24,28,60)(12,31,13,19)(14,37,40,50)(16,23,17,43)(18,44,47,27)(20,33,54,35)(21,41,57,42)(25,61,63,59)(26,46,32,68)(29,56,30,66)(34,53,36,52)(38,58,65,48)(39,51,71,55)(45,62,72,70)(49,69,67,64);(1,3,5,4)(2,6,7,8)(9,11,15,28)(10,18,22,47)(12,32,13,26)(14,38,40,65)(16,30,17,29)(19,49,31,67)(20,46,54,68)(21,56,57,66)(23,58,43,48)(24,50,60,37)(25,36,63,34)(27,41,44,42)(33,70,35,62)(39,64,71,69)(45,53,72,52)(51,59,55,61);(1,4,5,3)(2,6,7,8)(9,12,15,13)(10,19,22,31)(11,25,28,63)(14,33,40,35)(16,36,17,34)(18,45,47,72)(20,50,54,37)(21,51,57,55)(23,53,43,52)(24,59,60,61)(26,30,32,29)(27,62,44,70)(38,69,65,64)(39,42,71,41)(46,56,68,66)(48,67,58,49);(9,13,15,12)(10,20,22,54)(11,26,28,32)(14,39,40,71)(16,36,17,34)(18,46,47,68)(19,50,31,37)(21,53,57,52)(23,51,43,55)(24,49,60,67)(25,30,63,29)(27,62,44,70)(33,42,35,41)(38,64,65,69)(45,56,72,66)(48,61,58,59)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 411360);