#851467

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

Polycyclic presentation:

2^9|a^2=fh,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, 758663);