#6682940

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667604
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#36020	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2000 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)(5,6)(7,11)(8,10)(9,12)(13,14)(15,19)(16,21)(17,20)(18,22)(23,38)(24,32)(25,33)(26,31)(27,36)(28,34)(29,37)(30,35)(39,43)(40,42)(41,44);(5,7,8,12)(6,9,10,11)(13,15,17,25)(14,19,20,33)(16,27,28,24)(18,23,30,26)(21,36,34,32)(22,38,35,31)(29,40,41,39)(37,42,44,43);(1,2,3,4)(14,20)(15,23)(18,30)(19,31)(21,34)(24,39)(25,26)(27,40)(29,41)(32,42)(33,38)(36,43);(6,10)(9,11)(13,16)(14,21)(15,24)(17,28)(18,29)(19,32)(20,34)(22,37)(23,39)(25,27)(26,40)(30,41)(31,42)(33,36)(35,44)(38,43);(7,12)(9,11)(14,20)(15,23)(16,28)(18,30)(19,31)(24,40)(25,26)(27,39)(32,43)(33,38)(36,42)(37,44)

Polycyclic presentation:

2^9|b^2=fi,c^2=gi,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,d^b=di,e^b=efh,h^b=hi,h^c=hi,e^d=ei,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, 6590136);