#917168

order	512
nilpotency class	2
isoclinism	isoclinic class of #845297
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1978 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:

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

Polycyclic presentation:

2^9|a^2=fgh,b^2=fh,c^2=fghi,d^2=fh,e^2=ghi,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dg

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 824364);