#6683519

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667604
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 ◁ #2397 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2311 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6590715);