#691167

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	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2688 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:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=fh,c^2=gh,d^2=h,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, 598363);