#6432332

order	512
nilpotency class	3
isoclinism	isoclinic class of #23387
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂³	C₂³×D₄
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2438 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=gi,d^2=f,g^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=ei,g^c=gi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6339528);