#6684889

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682380
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

2055 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:

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

Polycyclic presentation:

2^9|a^2=fi,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,d^b=di,e^b=efh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6592085);