#696947

order	512
nilpotency class	2
isoclinism	isoclinic class of #686200
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂²×#121	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2764 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:

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

Polycyclic presentation:

2^9|b^2=gi,c^2=gh,d^2=fg,e^2=f,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, 604143);