#719550

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

2702 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)(6,7)(9,10,13,11)(12,14,16,15)(17,18)(19,24)(20,28)(21,26)(22,23)(25,30)(27,32)(29,31);(9,11,13,10)(12,14,16,15)(18,23)(21,31)(24,32)(25,28);(5,6,8,7)(9,12)(10,14)(11,15)(13,16)(17,19)(18,24)(20,29)(21,25)(22,27)(23,32)(26,30)(28,31);(1,2,3,4)(5,7,8,6)(17,20,22,30)(18,25,23,28)(19,26,27,29)(21,24,31,32);(17,21,22,31)(18,26,23,29)(19,25,27,28)(20,24,30,32)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=g,d^2=i,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, 626746);