#6761901

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682440
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	#37656	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1862 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

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, 6669097);