#6713569

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667604
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×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1562 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)(9,10,12,13)(11,14,15,16)(17,18)(19,24)(20,33)(21,26)(22,27)(23,28)(25,36)(29,45)(30,39)(31,41)(32,40)(34,48)(35,44)(37,43)(38,46)(42,47);(1,3)(2,5)(4,8)(6,7)(17,19,22,31)(18,24,27,41)(20,34,36,47)(21,32,23,30)(25,42,33,48)(26,40,28,39)(29,37,46,35)(38,44,45,43);(10,13)(14,16)(17,20)(18,25)(19,29)(21,35)(22,36)(23,37)(24,38)(26,43)(27,33)(28,44)(30,34)(31,46)(32,47)(39,42)(40,48)(41,45);(2,6)(5,7)(17,21,22,23)(18,26,27,28)(19,30,31,32)(20,35,36,37)(24,39,41,40)(25,43,33,44)(29,34,46,47)(38,42,45,48);(3,8)(5,7)(9,11,12,15)(10,14,13,16)(18,27)(19,30)(21,23)(24,40)(25,33)(29,34)(31,32)(35,37)(38,48)(39,41)(42,45)(46,47)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6620765);