#522209

order	512
nilpotency class	2
isoclinism	isoclinic class of #37644
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷

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

1938 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)(15,20)(16,18)(17,19);(1,3,4,2)(14,16)(19,20)(21,22,24,26)(23,27,28,25);(5,6)(7,10)(8,9)(11,12)(13,15,18,17)(14,19,16,20)(21,23,24,28)(22,25,26,27);(5,6)(7,10)(8,9)(11,12)(13,16,18,14)(15,20,17,19);(5,7,8,11)(6,10,9,12)(13,17,18,15)(14,20,16,19)

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=fh,d^2=f,e^2=g,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, 429405);