#6671827

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667932
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

2184 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 44 points:

(6,9)(7,11)(13,14)(15,21)(16,30)(17,23)(18,24)(19,34)(20,25)(22,31)(26,41)(27,36)(28,37)(29,39)(32,40)(33,38)(35,43)(42,44);(1,2)(3,4)(5,6,8,9)(7,10,11,12)(13,15)(14,21)(16,26)(17,27)(18,33)(19,28)(20,29)(22,35)(23,36)(24,38)(25,39)(30,41)(31,43)(32,42)(34,37)(40,44);(2,4)(13,16,20,17)(14,22,25,40)(15,26,29,27)(18,31,19,32)(21,35,39,44)(23,34,30,24)(28,42,33,43)(36,37,41,38);(5,7,8,11)(6,10,9,12)(13,17,20,16)(14,23,25,30)(15,27,29,26)(18,31,19,32)(21,36,39,41)(22,34,40,24)(28,42,33,43)(35,37,44,38);(2,4)(6,9)(10,12)(13,18,20,19)(14,24,25,34)(15,28,29,33)(16,31,17,32)(21,37,39,38)(22,23,40,30)(26,42,27,43)(35,41,44,36)

Polycyclic presentation:

2^9|b^2=f,c^2=i,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=efhi,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6579023);