#6676240

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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1802 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)(5,6,9,11)(7,12,8,10);(5,6)(7,10)(8,12)(9,11)(13,14,18,22)(15,23,26,33)(16,20,27,37)(17,29,31,21)(19,30,34,25)(24,38,39,44)(28,35,41,36)(32,42,43,40);(2,4)(13,15,18,26)(14,19,22,34)(16,24,27,39)(17,30,31,25)(20,32,37,43)(21,33,29,23)(28,42,41,40)(35,38,36,44);(5,7,9,8)(6,10,11,12)(13,16)(14,20)(15,24)(17,28)(18,27)(19,32)(21,36)(22,37)(23,38)(25,40)(26,39)(29,35)(30,42)(31,41)(33,44)(34,43);(5,8,9,7)(6,10,11,12)(14,21)(16,27)(17,31)(19,33)(20,35)(22,29)(23,34)(24,39)(25,30)(32,38)(36,37)(43,44)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=i,d^2=f,e^2=f,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, 6583436);