#7090526

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	C₂×#548	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1650 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)(17,18,23,21)(19,27,31,30)(20,33,35,25)(22,38,28,26)(24,39,37,32)(29,45,34,43)(36,42,44,47)(40,46,41,48);(9,10)(11,15)(12,14)(13,16)(17,19,28,39,23,31,22,32)(18,24,38,30,21,37,26,27)(20,34,44,46,35,29,36,48)(25,41,47,43,33,40,42,45);(1,3,5,4)(2,6,7,8)(9,11,12,16)(10,13,14,15)(17,20,23,35)(18,25,21,33)(19,29,31,34)(22,36,28,44)(24,40,37,41)(26,42,38,47)(27,43,30,45)(32,46,39,48);(11,16)(13,15)(17,21,23,18)(19,30,31,27)(20,25,35,33)(22,26,28,38)(24,32,37,39)(29,43,34,45)(36,47,44,42)(40,48,41,46);(1,4,5,3)(2,6,7,8)(10,14)(13,15)(18,21)(19,31)(20,35)(26,38)(32,39)(36,44)(40,41)(43,45)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=ghi,d^2=i,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,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, 6997722);