#6669923

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667964
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#3489	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1657 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)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,18)(19,31)(20,36)(21,30)(22,28)(23,29)(24,27)(25,34)(26,38)(32,43)(33,48)(35,44)(37,46)(39,45)(40,42)(41,47);(1,3,5,11)(2,6,8,9)(4,10,12,16)(7,13,14,15)(17,19)(18,25)(20,37)(21,32)(22,39)(23,34)(24,35)(26,28)(27,43)(29,31)(30,44)(33,40)(36,42)(38,41)(45,47)(46,48);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(17,20)(18,26)(19,22)(21,38)(23,33)(24,39)(25,42)(27,36)(28,44)(29,45)(30,48)(31,46)(32,37)(34,41)(35,40)(43,47);(2,8)(6,9)(7,14)(13,15)(17,21,23,24)(18,27,29,30)(19,32,34,35)(20,38,33,39)(22,37,41,40)(25,43,31,44)(26,36,45,48)(28,42,47,46);(3,11)(6,9)(10,16)(13,15)(17,22,23,41)(18,28,29,47)(19,33,34,20)(21,40,24,37)(25,36,31,48)(26,44,45,43)(27,46,30,42)(32,38,35,39)

Polycyclic presentation:

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

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, 6577119);