#6692530

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667556
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ 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)(3,7)(4,6)(5,8)(9,10,13,15)(11,14,12,16)(17,18,22,27)(19,29,32,24)(20,25,23,28)(21,26,35,41)(30,42,36,37)(31,43,46,38)(33,39,34,40)(44,48,45,47);(1,3)(2,5)(4,8)(6,7)(9,11,13,12)(10,14,15,16)(19,30)(20,23)(21,33)(24,37)(25,28)(26,39)(29,42)(32,36)(34,35)(40,41)(44,45)(47,48);(9,12,13,11)(10,14,15,16)(17,19)(18,24)(20,30)(21,31)(22,32)(23,36)(25,37)(26,38)(27,29)(28,42)(33,44)(34,45)(35,46)(39,47)(40,48)(41,43);(2,6)(5,7)(17,20,22,23)(18,25,27,28)(19,30,32,36)(21,34,35,33)(24,37,29,42)(26,40,41,39)(31,45,46,44)(38,48,43,47);(3,8)(5,7)(17,21,22,35)(18,26,27,41)(19,31,32,46)(20,33,23,34)(24,38,29,43)(25,39,28,40)(30,44,36,45)(37,47,42,48)

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=gi,d^2=i,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,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

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6599726);