#7004426

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
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	#2388	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1706 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:

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

Polycyclic presentation:

2^9|a^2=fgi,c^2=f,d^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eh,d^c=dh,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6911622);