#6874238

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667932
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	C₂×#121	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1596 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:

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

Polycyclic presentation:

2^9|a^2=fgi,c^2=i,d^2=fh,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=dfi,c^b=ch,e^b=efh,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, 6781434);