#7090889

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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1570 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)(3,8)(4,6)(5,7)(9,10)(11,13)(12,14)(15,16)(17,18,23,28,24,29,22,38)(19,26,31,41,20,32,34,25)(21,27,36,44,37,45,39,43)(30,42,35,48,33,47,46,40);(9,10)(11,15)(12,14)(13,16)(17,19,24,20)(18,25,29,41)(21,35,37,46)(22,31,23,34)(26,28,32,38)(27,42,45,47)(30,39,33,36)(40,43,48,44);(1,3,5,4)(2,6,7,8)(9,11,12,16)(10,13,14,15)(17,20,24,19)(18,26,29,32)(21,33,37,30)(22,31,23,34)(25,28,41,38)(27,42,45,47)(35,36,46,39)(40,44,48,43);(11,16)(13,15)(17,21,22,39,24,37,23,36)(18,27,38,43,29,45,28,44)(19,30,34,46,20,33,31,35)(25,40,32,47,41,48,26,42);(1,4,5,3)(2,6,7,8)(10,14)(13,15)(17,22,24,23)(18,28,29,38)(19,31,20,34)(21,36,37,39)(25,32,41,26)(27,43,45,44)(30,46,33,35)(40,42,48,47)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=gh,d^2=fi,e^2=g,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, 6998085);