#521096

order	512
nilpotency class	2
isoclinism	isoclinic class of #2668
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

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

2110 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=hi,d^2=g,e^2=f,b^a=bf,c^a=cg,d^a=dfg,c^b=ch,d^b=df

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 428292);