#7102020

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

1253 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)(19,25)(20,26)(21,27)(22,24)(23,28)(29,38)(30,36)(31,40)(32,42)(33,41)(34,39)(35,37)(43,48)(44,46)(45,47);(1,3,5,4)(2,6,8,7)(9,10)(11,15)(12,14)(13,16)(18,24)(20,32)(21,22)(25,36)(26,41)(27,28)(29,43)(30,34)(33,35)(38,47)(39,40)(44,45);(1,3,5,4)(2,7,8,6)(9,11,12,16)(10,13,14,15)(17,19,23,31)(18,25,28,40)(20,29,33,45)(21,34,22,30)(24,36,27,39)(26,38,41,47)(32,43,35,44)(37,46,42,48);(1,3,5,4)(2,6,8,7)(11,16)(13,15)(17,20)(18,26)(19,29)(21,32)(22,35)(23,33)(24,37)(25,38)(27,42)(28,41)(30,44)(31,45)(34,43)(36,46)(39,48)(40,47);(1,4,5,3)(2,7,8,6)(10,14)(13,15)(17,21,23,22)(18,27,28,24)(19,30,31,34)(20,32,33,35)(25,36,40,39)(26,42,41,37)(29,44,45,43)(38,46,47,48)

Polycyclic presentation:

2^9|b^2=g,c^2=ghi,d^2=g,e^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,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, 7009216);