#7102025

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1249 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)(3,6)(4,7)(5,8)(9,10)(11,13)(12,14)(15,16)(17,18)(19,29)(20,26)(21,27)(22,24)(23,28)(25,32)(30,43)(31,40)(33,42)(34,41)(35,37)(36,38)(39,46)(44,47)(45,48);(1,3,5,4)(2,6,8,7)(9,10)(11,15)(12,14)(13,16)(18,24)(20,33)(21,22)(25,37)(26,41)(27,28)(29,40)(30,44)(31,35)(34,36)(39,43)(45,46);(1,3,5,4)(2,7,8,6)(9,11,12,16)(10,13,14,15)(17,19,23,32)(18,25,28,29)(20,30,34,46)(21,35,22,31)(24,37,27,40)(26,39,41,43)(33,44,36,45)(38,47,42,48);(1,3,5,4)(2,6,8,7)(11,16)(13,15)(17,20)(18,26)(19,30)(21,33)(22,36)(23,34)(24,38)(25,39)(27,42)(28,41)(29,43)(31,45)(32,46)(35,44)(37,47)(40,48);(1,4,5,3)(2,7,8,6)(10,14)(13,15)(17,21,23,22)(18,27,28,24)(19,31,32,35)(20,33,34,36)(25,37,29,40)(26,42,41,38)(30,45,46,44)(39,47,43,48)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7009221);