#6694389

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1316 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=g,c^2=g,d^2=f,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dfi,c^b=ch,e^b=efh,h^b=hi,d^c=di,h^c=hi,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, 6601585);