#7094533

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1260 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)(3,6,4,7)(9,10)(11,13)(12,14)(15,16)(17,18)(19,24)(20,25)(21,23)(22,27)(26,34)(28,37)(29,38)(30,35)(31,36)(32,41)(33,40)(39,45)(42,46)(43,47)(44,48);(1,3,5,4)(2,6,8,7)(9,10)(11,15)(12,14)(13,16)(18,23)(20,31)(21,34)(24,35)(26,27)(28,42)(30,45)(32,33)(36,41)(38,39)(43,44)(46,48);(1,3,5,4)(2,7,8,6)(9,11,12,16)(10,13,14,15)(17,19,22,29)(18,24,27,38)(20,28,33,43)(21,30,34,45)(23,35,26,39)(25,37,40,47)(31,42,32,44)(36,46,41,48);(11,16)(13,15)(17,20,22,33)(18,25,27,40)(19,28,29,43)(21,31,34,32)(23,36,26,41)(24,37,38,47)(30,42,45,44)(35,46,39,48);(1,4,5,3)(2,7,8,6)(10,14)(13,15)(19,29)(24,38)(28,43)(30,45)(35,39)(37,47)(42,44)(46,48)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=ghi,d^2=i,e^2=g,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, 7001729);