#7089817

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	#2407	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1434 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)(17,18,24,29)(19,26,20,32)(21,27,37,46)(22,28,23,38)(25,34,41,31)(30,42,33,43)(35,48,47,40)(36,44,39,45);(9,10)(11,15)(12,14)(13,16)(17,19,24,20)(18,25,29,41)(21,35,37,47)(22,31,23,34)(26,28,32,38)(27,43,46,42)(30,39,33,36)(40,44,48,45);(1,3,5,4)(2,6,7,8)(9,11,12,16)(10,13,14,15)(17,20,24,19)(18,26,29,32)(21,30,37,33)(22,31,23,34)(25,28,41,38)(27,43,46,42)(35,39,47,36)(40,45,48,44);(11,16)(13,15)(17,21,24,37)(18,27,29,46)(19,30,20,33)(22,36,23,39)(25,40,41,48)(26,42,32,43)(28,44,38,45)(31,47,34,35);(1,4,5,3)(2,6,7,8)(10,14)(13,15)(17,22,24,23)(18,28,29,38)(19,31,20,34)(21,36,37,39)(25,32,41,26)(27,44,46,45)(30,47,33,35)(40,43,48,42)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=gh,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=df,e^b=eh,f^b=fi,d^c=dhi,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, 6997013);