#6692947

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667772
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×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1566 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)(3,7)(4,6)(5,8)(9,10,13,15)(11,14,12,16)(18,24)(25,38)(26,36)(27,39)(28,40)(33,42)(41,45)(43,48);(1,3,4,8)(2,5,6,7)(9,11,13,12)(10,14,15,16)(17,18)(19,28)(20,26)(21,33)(22,24)(23,36)(25,29)(27,34)(30,45)(31,40)(32,39)(35,42)(37,38)(41,47)(43,44)(46,48);(9,12,13,11)(10,14,15,16)(17,19,22,31)(18,25,24,38)(20,29,23,37)(21,30,35,47)(26,40,36,28)(27,41,39,45)(32,44,34,46)(33,48,42,43);(2,6)(5,7)(17,20,22,23)(18,26,24,36)(19,29,31,37)(21,34,35,32)(25,40,38,28)(27,42,39,33)(30,46,47,44)(41,43,45,48);(3,8)(5,7)(17,21,22,35)(18,27,24,39)(19,30,31,47)(20,32,23,34)(25,41,38,45)(26,33,36,42)(28,43,40,48)(29,44,37,46)

Polycyclic presentation:

2^9|a^2=g,b^2=fgi,c^2=gi,d^2=i,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfi,c^b=ch,e^b=efh,h^b=hi,h^c=hi,e^d=ei,h^e=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6600143);