#7080715

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996355
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	#25125	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1618 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)(3,7)(4,6)(5,8)(9,10)(11,13)(12,14)(15,16)(17,18,22,27,23,28,36,38)(19,29,31,26,32,42,21,24)(20,25,35,41,34,40,46,47)(30,43,45,39,44,48,33,37);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,14)(13,16)(17,19,23,32)(18,24,28,26)(20,30,34,44)(21,36,31,22)(25,37,40,39)(27,42,38,29)(33,46,45,35)(41,48,47,43);(2,6)(5,7)(9,11,12,16)(10,13,14,15)(17,20,23,34)(18,25,28,40)(19,30,32,44)(21,33,31,45)(22,35,36,46)(24,37,26,39)(27,41,38,47)(29,43,42,48);(3,8)(5,7)(11,16)(13,15)(17,21,22,19,23,31,36,32)(18,26,27,42,28,24,38,29)(20,33,35,30,34,45,46,44)(25,39,41,48,40,37,47,43);(10,14)(13,15)(20,34)(25,40)(30,44)(33,45)(35,46)(37,39)(41,47)(43,48)

Polycyclic presentation:

2^9|a^2=fg,b^2=i,c^2=hi,d^2=fg,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=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, 6987911);