#625759

order	512
nilpotency class	2
isoclinism	isoclinic class of #616960
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1919 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=fghi,c^2=fgi,d^2=fg,e^2=fh,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, 532955);