#528313

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

3274 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)(15,21,23,19)(16,24,25,20)(22,26,28,27);(10,12)(14,18)(19,21)(20,24)(26,27);(1,2,4,3)(5,6)(7,8)(9,10)(11,12)(13,15,17,23)(14,19,18,21)(16,22,25,28)(20,26,24,27);(5,6)(7,8)(13,14)(15,19)(16,20)(17,18)(21,23)(22,26)(24,25)(27,28);(1,3,4,2)(13,16)(14,20)(15,22)(17,25)(18,24)(19,26)(21,27)(23,28)

Polycyclic presentation:

2^9|a^2=i,c^2=fgi,e^2=gi,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, 435509);