#7532680

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514040
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	#45255	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1514 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)(3,8)(4,6)(5,7)(9,10)(11,15)(12,14)(13,16)(17,18,23,19)(20,24,30,26)(21,27,22,25)(28,31,32,29);(1,3,5,4)(2,6,7,8)(11,16)(13,15)(17,18,23,19)(20,28,30,32)(21,25,22,27)(24,31,26,29);(9,11,12,16)(10,13,14,15)(17,19,23,18)(20,24,30,26)(21,25,22,27)(28,31,32,29);(1,4,5,3)(2,6,7,8)(10,14)(13,15)(17,20)(18,24)(19,26)(21,29)(22,31)(23,30)(25,32)(27,28);(2,7)(6,8)(17,21,23,22)(18,25,19,27)(20,29,30,31)(24,32,26,28)

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7439876);