#857208

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

2154 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fh,c^2=ghi,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, 764404);