#7101204

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	#31066	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1431 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:

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

Polycyclic presentation:

2^9|a^2=gi,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=dfi,e^b=eh,f^b=fi,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, 7008400);