#6684877

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682440
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2142 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6592073);