#7081617

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×#546	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1560 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)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18,22,26)(19,27,31,38)(20,33,34,24)(21,36,28,25)(23,30,39,32)(29,43,46,41)(35,40,42,48)(37,45,47,44);(2,5)(3,7)(9,10)(11,15)(12,14)(13,16)(17,19,28,30,22,31,21,32)(18,23,36,38,26,39,25,27)(20,29,42,44,34,46,35,45)(24,37,48,43,33,47,40,41);(2,5)(6,8)(9,11,12,16)(10,13,14,15)(17,20)(18,24)(19,29)(21,35)(22,34)(23,37)(25,40)(26,33)(27,41)(28,42)(30,44)(31,46)(32,45)(36,48)(38,43)(39,47);(1,3,4,7)(2,6,5,8)(11,16)(13,15)(19,30)(21,28)(23,38)(25,36)(27,39)(29,44)(31,32)(35,42)(37,43)(40,48)(41,47)(45,46);(10,14)(13,15)(19,31)(20,34)(23,39)(24,33)(27,38)(30,32)(35,42)(40,48)

Polycyclic presentation:

2^9|a^2=i,b^2=fg,c^2=h,d^2=gi,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=df,e^b=ehi,d^c=dh,e^c=ei,f^d=fi

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, 6988813);