#851466

order	512
nilpotency class	2
isoclinism	isoclinic class of #845297
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂⁶	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3118 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)(3,8)(4,7)(5,6)(10,14)(11,12)(18,21)(20,23)(25,26,28,30)(27,29,31,32);(2,6)(7,8)(9,10,13,14)(11,15,12,16)(17,18,22,21)(19,23,24,20)(27,31)(29,32);(1,3,5,4)(2,7,6,8)(9,11,13,12)(10,15,14,16)(17,19,22,24)(18,23,21,20)(25,27)(26,29)(28,31)(30,32);(1,2)(3,7)(4,8)(5,6)(9,10)(11,15)(12,16)(13,14)(17,20)(18,19)(21,24)(22,23)(25,26)(27,29)(28,30)(31,32);(1,4,5,3)(2,8,6,7)(9,12,13,11)(10,16,14,15)(17,21,22,18)(19,20,24,23)

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=f,e^2=f,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, 758662);