#624390

order	512
nilpotency class	2
isoclinism	isoclinic class of #616960
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

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

4510 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:

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

Polycyclic presentation:

2^9|b^2=fi,c^2=hi,d^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, 531586);