#6676386

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667932
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	#2540	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1822 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=i,d^2=fi,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efhi,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6583582);