#696696

order	512
nilpotency class	2
isoclinism	isoclinic class of #686200
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂×#392	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2258 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,d^2=fg,e^2=fg,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef,d^c=df

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, 603892);