#6684833

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682360
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

2089 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=fg,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,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, 6592029);