#719712

order	512
nilpotency class	2
isoclinism	isoclinic class of #686200
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

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

2430 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=gh,d^2=fi,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, 626908);