#524713

order	512
nilpotency class	2
isoclinism	isoclinic class of #37644
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, prime radical, cosocle	C₂⁴	C₂⁵
commutator	C₂³	C₂⁴×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3514 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fh,c^2=fgi,d^2=ghi,e^2=f,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=ch,d^b=df

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 431909);