#515172

order	512
nilpotency class	2
isoclinism	isoclinic class of #2613
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

2190 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)(6,9)(10,12)(13,14)(15,18)(16,17)(19,20);(5,6,8,9)(7,10,11,12)(21,22,24,26)(23,27,28,25);(7,11)(10,12)(14,17)(18,20)(21,23,24,28)(22,25,26,27);(1,3,4,2)(5,7,8,11)(6,10,9,12);(13,15,16,19)(14,18,17,20)

Polycyclic presentation:

2^9|a^2=i,b^2=fh,c^2=h,d^2=fi,e^2=g,b^a=bf,c^a=cg,c^b=ch,d^c=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, 422368);