#6676856

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	#392	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1846 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);(5,7,10,6)(8,12,9,11)(13,14,18,23)(15,24,27,37)(16,20,28,35)(17,21,31,22)(19,30,33,26)(25,38,39,43)(29,36,41,34)(32,42,44,40);(2,4)(13,15,18,27)(14,19,23,33)(16,25,28,39)(17,30,31,26)(20,32,35,44)(21,24,22,37)(29,42,41,40)(34,43,36,38);(5,8,10,9)(6,11,7,12)(13,16)(14,20)(15,25)(17,29)(18,28)(19,32)(21,36)(22,34)(23,35)(24,38)(26,40)(27,39)(30,42)(31,41)(33,44)(37,43);(5,9,10,8)(6,11,7,12)(14,21)(16,28)(17,31)(19,24)(20,34)(22,23)(25,39)(26,30)(32,43)(33,37)(35,36)(38,44)

Polycyclic presentation:

2^9|b^2=fi,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=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, 6584052);