#863409

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, prime radical, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1760 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=ghi,d^2=g,e^2=fgh,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, 770605);