#6676899

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	#42800	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1861 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)(7,10)(8,12)(9,11)(13,14,18,16)(15,23,27,20)(17,29,31,21)(19,22,33,30)(24,28,35,26)(25,34,40,37)(32,42,43,36)(38,41,44,39);(5,7,10,6)(8,12,9,11)(13,15,18,27)(14,20,16,23)(17,25,31,40)(19,26,33,28)(21,34,29,37)(22,35,30,24)(32,41,43,39)(36,44,42,38);(2,4)(13,16,18,14)(15,24,27,35)(17,29,31,21)(19,22,33,30)(20,26,23,28)(25,38,40,44)(32,36,43,42)(34,41,37,39);(5,8,10,9)(6,11,7,12)(13,17)(14,21)(15,25)(16,29)(18,31)(19,32)(20,34)(22,36)(23,37)(24,38)(26,41)(27,40)(28,39)(30,42)(33,43)(35,44);(5,9,10,8)(6,11,7,12)(15,26)(17,31)(19,33)(20,35)(21,29)(22,30)(23,24)(25,39)(27,28)(34,38)(37,44)(40,41)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=i,d^2=fi,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efhi,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, 6584095);