#7086154

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996639
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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2401 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1383 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=i,c^2=h,d^2=fi,e^2=gi,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,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, 6993350);