#6443466

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438975
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2018 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6350662);