#6720143

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	C₂²×D₈	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1415 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=gi,c^2=f,d^2=fi,e^2=gi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efhi,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, 6627339);