#6631346

order	512
nilpotency class	3
isoclinism	isoclinic class of #6631116
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1534 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2,5,4)(3,7,8,6)(9,10)(11,13)(12,14)(15,16)(17,18,21,23)(19,27,29,42)(20,25,32,38)(22,35,34,26)(24,36,37,30)(28,43,31,45)(33,48,47,41)(39,46,40,44);(1,3,5,8)(2,6,4,7)(11,15)(13,16)(18,23)(24,37)(25,38)(26,35)(27,42)(39,40)(41,48)(43,45);(2,4)(6,7)(9,11)(10,13)(12,15)(14,16)(17,19,21,29)(18,24,23,37)(20,31,32,28)(22,36,34,30)(25,40,38,39)(26,42,35,27)(33,44,47,46)(41,43,48,45);(1,4,5,2)(3,7,8,6)(10,14)(11,15)(17,20,21,32)(18,25,23,38)(19,28,29,31)(22,33,34,47)(24,39,37,40)(26,41,35,48)(27,43,42,45)(30,44,36,46);(18,23)(20,32)(24,37)(26,35)(27,42)(28,31)(33,47)(44,46)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6538542);