#7101016

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	#2414	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1405 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=gh,d^2=g,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=ehi,f^b=fi,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, 7008212);