#6874214

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667900
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	#38032	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1621 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)(17,18,23,36,22,27,19,24)(20,30,28,25,33,37,32,40)(21,26,34,41,35,42,29,38)(31,44,43,39,46,47,45,48);(1,2)(3,6)(4,8)(5,7)(9,11)(10,13)(12,16)(14,15)(17,19,22,23)(18,24,27,36)(20,28,33,32)(21,29,35,34)(25,37,40,30)(26,38,42,41)(31,43,46,45)(39,47,48,44);(11,16)(13,15)(17,20,22,33)(18,25,27,40)(19,28,23,32)(21,31,35,46)(24,37,36,30)(26,39,42,48)(29,43,34,45)(38,47,41,44);(1,3,5,4)(2,6,7,8)(10,14)(13,15)(17,21)(18,26)(19,29)(20,31)(22,35)(23,34)(24,38)(25,39)(27,42)(28,43)(30,44)(32,45)(33,46)(36,41)(37,47)(40,48);(1,4,5,3)(2,6,7,8)(17,19,22,23)(18,24,27,36)(20,28,33,32)(21,34,35,29)(25,37,40,30)(26,41,42,38)(31,45,46,43)(39,44,48,47)

Polycyclic presentation:

2^9|a^2=fgi,b^2=i,c^2=i,d^2=fh,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,g^c=gi,e^d=ei

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6781410);