#6811586

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667980
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 ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2109 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6718782);