#631785

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	C₂×#392	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2091 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=gi,c^2=gi,d^2=fgi,e^2=fgh,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, 538981);