#6999581

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994139
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×#548	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1635 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:

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

Polycyclic presentation:

2^9|b^2=fi,c^2=i,d^2=i,e^2=f,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dh,e^c=ei,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, 6906777);