#7090226

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
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×#550	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1669 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)(19,27,30,41)(20,31,32,25)(22,35,36,26)(24,37,34,29)(28,42,43,40)(33,39,47,46)(38,44,45,48);(9,10)(11,15)(12,14)(13,16)(17,19,22,29,23,30,36,37)(18,24,26,27,21,34,35,41)(20,28,33,44,32,43,47,48)(25,38,39,40,31,45,46,42);(1,3,5,4)(2,6,7,8)(9,11,12,16)(10,13,14,15)(17,20,23,32)(18,25,21,31)(19,28,30,43)(22,33,36,47)(24,38,34,45)(26,39,35,46)(27,40,41,42)(29,44,37,48);(11,16)(13,15)(17,21,23,18)(19,27,30,41)(20,31,32,25)(22,26,36,35)(24,37,34,29)(28,40,43,42)(33,39,47,46)(38,48,45,44);(1,4,5,3)(2,6,7,8)(10,14)(13,15)(20,32)(25,31)(28,43)(33,47)(38,45)(39,46)(40,42)(44,48)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=gh,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=df,e^b=eh,d^c=dh,e^c=ei,f^d=fi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6997422);