#6692570

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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1425 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)(19,25)(20,26)(21,27)(22,38)(23,29)(24,30)(28,31)(32,41)(33,45)(34,42)(35,43)(36,44)(37,46)(39,48)(40,47);(1,3)(2,5)(4,8)(6,7)(9,11,13,12)(10,14,15,16)(17,19)(18,25)(20,35)(21,32)(22,39)(23,40)(24,34)(26,43)(27,41)(28,44)(29,47)(30,42)(31,36)(33,37)(38,48)(45,46);(9,12,13,11)(10,14,15,16)(17,20,24,33)(18,26,30,45)(19,31,34,22)(21,36,23,39)(25,28,42,38)(27,44,29,48)(32,37,40,35)(41,46,47,43);(2,6)(5,7)(17,21,24,23)(18,27,30,29)(19,32,34,40)(20,36,33,39)(22,35,31,37)(25,41,42,47)(26,44,45,48)(28,46,38,43);(3,8)(5,7)(17,22,24,31)(18,28,30,38)(19,33,34,20)(21,37,23,35)(25,26,42,45)(27,43,29,46)(32,36,40,39)(41,48,47,44)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=gi,d^2=i,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,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, 6599766);