#6692607

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

1420 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)(17,18,23,28)(19,24,31,39)(20,25,22,27)(21,33,35,26)(29,37,36,42)(30,44,46,38)(32,41,34,40)(43,48,45,47);(1,3)(2,5)(4,8)(6,7)(9,11,13,12)(10,14,15,16)(19,29)(20,22)(21,32)(24,37)(25,27)(26,40)(31,36)(33,41)(34,35)(39,42)(43,45)(47,48);(9,12,13,11)(10,14,15,16)(17,19)(18,24)(20,29)(21,30)(22,36)(23,31)(25,37)(26,38)(27,42)(28,39)(32,43)(33,44)(34,45)(35,46)(40,47)(41,48);(2,6)(5,7)(17,20,23,22)(18,25,28,27)(19,29,31,36)(21,34,35,32)(24,37,39,42)(26,41,33,40)(30,45,46,43)(38,48,44,47);(3,8)(5,7)(17,21,23,35)(18,26,28,33)(19,30,31,46)(20,32,22,34)(24,38,39,44)(25,40,27,41)(29,43,36,45)(37,47,42,48)

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=g,d^2=i,e^2=i,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, 6599803);