#7617476

order	512
nilpotency class	3
isoclinism	isoclinic class of #7519008
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1251 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

(1,2)(3,7)(4,8)(5,6)(9,10,14,11)(12,16,13,15)(17,18)(19,21)(20,23)(22,27)(24,26)(25,29)(28,32)(30,31);(2,6)(7,8)(9,11,14,10)(12,15,13,16)(17,19,22,26)(18,24,27,21)(20,31,29,28)(23,32,25,30);(1,3,5,4)(2,7,6,8)(9,12,14,13)(10,15,11,16)(17,20)(18,25)(19,28)(21,30)(22,29)(23,27)(24,32)(26,31);(9,13,14,12)(10,16,11,15)(17,21,23,31,22,24,25,28)(18,26,20,32,27,19,29,30);(1,4,5,3)(2,7,6,8)(18,27)(20,29)(21,24)(28,31)

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=fg,d^2=h,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,d^c=di,e^c=ei,e^d=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7524672);