#7146693

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, prime radical	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1330 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

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

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, 7053889);