#7086282

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

1380 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)(17,18,23,29,24,30,22,28)(19,31,34,43,35,42,33,25)(20,26,37,45,38,46,40,41)(21,27,32,47,39,44,36,48);(9,10)(11,15)(12,14)(13,16)(17,19,24,35)(18,25,30,43)(20,36,38,32)(21,37,39,40)(22,34,23,33)(26,44,46,27)(28,31,29,42)(41,48,45,47);(2,5)(6,8)(9,11,12,16)(10,13,14,15)(17,20,24,38)(18,26,30,46)(19,32,35,36)(21,34,39,33)(22,40,23,37)(25,27,43,44)(28,41,29,45)(31,47,42,48);(11,16)(13,15)(17,21,22,36,24,39,23,32)(18,27,28,48,30,44,29,47)(19,20,33,40,35,38,34,37)(25,41,42,46,43,45,31,26);(1,3,4,7)(2,6,5,8)(10,14)(13,15)(17,22,24,23)(18,28,30,29)(19,33,35,34)(20,37,38,40)(21,32,39,36)(25,42,43,31)(26,45,46,41)(27,47,44,48)

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=ehi,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, 6993478);