#625814

order	512
nilpotency class	2
isoclinism	isoclinic class of #616960
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

2422 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)(9,10,14,11)(12,16,13,15)(17,18)(19,20)(21,29)(22,23)(24,27)(25,30)(26,32)(28,31);(6,8)(9,11,14,10)(12,16,13,15)(18,23)(20,27)(25,29)(28,32);(5,6)(7,8)(9,12,14,13)(10,15,11,16)(17,19,22,24)(18,20,23,27)(21,26,30,31)(25,28,29,32);(1,2,3,4)(9,13,14,12)(10,16,11,15)(17,20)(18,24)(19,23)(21,28)(22,27)(25,31)(26,29)(30,32);(17,21,22,30)(18,25,23,29)(19,26,24,31)(20,28,27,32)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=fg,d^2=fi,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=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, 533010);