#6448180

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437521
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	C₂³×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#533 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1662 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=fg,c^2=fgi,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,h^b=hi,h^c=hi,h^d=hi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6355376);