#7078652

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	#2414	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1412 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)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18,24,29)(19,30,35,42)(20,36,21,26)(22,28,23,27)(25,33,43,34)(31,41,37,47)(32,44,48,40)(38,45,39,46);(9,10)(11,15)(12,14)(13,16)(17,19,24,35)(18,25,29,43)(20,37,21,31)(22,34,23,33)(26,44,36,40)(27,30,28,42)(32,39,48,38)(41,45,47,46);(2,5)(6,8)(9,11,12,16)(10,13,14,15)(17,20)(18,26)(19,31)(21,24)(22,39)(23,38)(25,40)(27,46)(28,45)(29,36)(30,47)(32,34)(33,48)(35,37)(41,42)(43,44);(1,3,4,7)(2,6,5,8)(11,16)(13,15)(17,21,24,20)(18,26,29,36)(19,32,35,48)(22,39,23,38)(25,41,43,47)(27,45,28,46)(30,44,42,40)(31,33,37,34);(10,14)(13,15)(17,22,24,23)(18,27,29,28)(19,33,35,34)(20,38,21,39)(25,42,43,30)(26,45,36,46)(31,32,37,48)(40,47,44,41)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=hi,d^2=g,e^2=i,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, 6985848);