#7532382

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	#45263	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1820 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=i,c^2=g,d^2=fi,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, 7439578);