#6883655

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682440
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 ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1222 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=fi,c^2=gi,d^2=fhi,e^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,d^c=di,e^c=ei,e^d=ei

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, 6790851);