#6692711

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667580
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)(19,24)(20,31)(21,26)(22,25)(23,27)(28,36)(29,38)(30,39)(32,41)(33,40)(34,42)(35,37)(43,46)(44,47)(45,48);(1,3)(2,5)(4,8)(6,7)(9,11,13,12)(10,14,15,16)(19,28)(20,22)(21,32)(24,36)(25,31)(26,41)(30,35)(33,34)(37,39)(40,42)(43,44)(46,47);(9,12,13,11)(10,14,15,16)(17,19)(18,24)(20,28)(21,29)(22,35)(23,30)(25,37)(26,38)(27,39)(31,36)(32,43)(33,44)(34,45)(40,47)(41,46)(42,48);(2,6)(5,7)(17,20,23,22)(18,25,27,31)(19,28,30,35)(21,33,34,32)(24,37,39,36)(26,41,42,40)(29,44,45,43)(38,46,48,47);(3,8)(5,7)(17,21,23,34)(18,26,27,42)(19,29,30,45)(20,32,22,33)(24,38,39,48)(25,40,31,41)(28,43,35,44)(36,46,37,47)

Polycyclic presentation:

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

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6599907);