#7090653

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	#24425	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1670 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)(3,8)(4,6)(5,7)(9,10)(11,13)(12,14)(15,16)(17,18,23,27)(19,29,34,46)(20,25,21,26)(22,38,30,28)(24,39,42,33)(31,41,35,48)(32,40,47,43)(36,45,37,44);(9,10)(11,15)(12,14)(13,16)(17,19,30,39,23,34,22,33)(18,24,38,46,27,42,28,29)(20,35,37,32,21,31,36,47)(25,43,45,41,26,40,44,48);(1,3,5,4)(2,6,7,8)(9,11,12,16)(10,13,14,15)(17,20)(18,25)(19,31)(21,23)(22,36)(24,40)(26,27)(28,44)(29,41)(30,37)(32,33)(34,35)(38,45)(39,47)(42,43)(46,48);(11,16)(13,15)(17,21,23,20)(18,26,27,25)(19,32,34,47)(22,36,30,37)(24,41,42,48)(28,44,38,45)(29,40,46,43)(31,39,35,33);(1,4,5,3)(2,6,7,8)(10,14)(13,15)(18,27)(20,21)(24,42)(28,38)(29,46)(31,35)(32,47)(36,37)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,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=ehi,d^c=dh,e^c=ei,e^d=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, 6997849);