#6806884

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806839
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	#42995	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2019 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:

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

Polycyclic presentation:

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

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, 6714080);