#522198

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

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

2290 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)(5,6,8,7)(17,18,21,22)(19,25,26,23)(20,28,29,24)(27,30,32,31);(5,7,8,6)(9,10,12,14)(11,15,16,13)(18,22)(19,26)(24,28)(27,32);(1,2,4,3)(9,11,12,16)(10,13,14,15)(17,19,21,26)(18,23,22,25)(20,27,29,32)(24,30,28,31);(1,3,4,2)(17,18)(19,23)(20,24)(21,22)(25,26)(27,30)(28,29)(31,32);(17,20,21,29)(18,24,22,28)(19,27,26,32)(23,30,25,31)

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=fgh,d^2=fg,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, 429394);