#6682821

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	#36009	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2048 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=g,d^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,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, 6590017);