#523239

order	512
nilpotency class	2
isoclinism	isoclinic class of #37644
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂⁴	C₂⁵
commutator	C₂³	C₂⁴×C₄
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3802 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=i,d^2=fgi,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, 430435);