#6671027

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

1779 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)(5,6,8,9)(7,11,12,10);(5,6)(7,10)(8,9)(11,12)(13,14,19,23)(15,24,27,34)(16,21,18,32)(17,29,31,37)(20,33,36,25)(22,30,38,28)(26,35,42,44)(39,40,43,41);(2,4)(6,9)(10,11)(13,15,19,27)(14,20,23,36)(16,25,18,33)(17,26,31,42)(21,34,32,24)(22,35,38,44)(28,40,30,41)(29,39,37,43);(5,7,8,12)(6,10,9,11)(13,16,19,18)(14,21,23,32)(15,25,27,33)(17,30,31,28)(20,34,36,24)(22,29,38,37)(26,41,42,40)(35,39,44,43);(13,17,19,31)(14,22,23,38)(15,26,27,42)(16,28,18,30)(20,35,36,44)(21,37,32,29)(24,39,34,43)(25,40,33,41)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=i,d^2=fi,e^2=i,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, 6578223);