#6682284

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667772
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

2220 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)(23,36)(24,31)(25,32)(26,30)(27,33)(28,35)(29,34)(37,42)(38,41)(39,43)(40,44);(5,7)(6,9)(8,12)(10,11)(13,15,17,25)(14,19,20,32)(16,24,27,39)(18,23,29,26)(21,31,33,43)(22,36,34,30)(28,37,40,38)(35,42,44,41);(1,2,3,4)(14,20)(15,23)(18,29)(19,30)(21,33)(24,37)(25,26)(28,40)(31,41)(32,36)(38,39)(42,43);(6,10)(9,11)(13,16)(14,21)(15,24)(17,27)(18,28)(19,31)(20,33)(22,35)(23,37)(25,39)(26,38)(29,40)(30,41)(32,43)(34,44)(36,42);(7,12)(9,11)(14,20)(15,23)(16,27)(18,29)(19,30)(24,38)(25,26)(31,42)(32,36)(35,44)(37,39)(41,43)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6589480);