#6485669

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470185
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	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1413 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)(3,6)(4,5)(7,8)(9,10)(11,15)(12,16)(13,14)(17,18,24,29)(19,30,34,42)(20,35,21,27)(22,28,23,38)(25,39,43,33)(26,37,44,36)(31,40,32,46)(41,48,45,47);(2,5)(6,8)(9,10)(11,16)(12,15)(13,14)(17,19)(18,25)(20,31)(21,32)(22,33)(23,39)(24,34)(26,40)(27,41)(28,42)(29,43)(30,38)(35,45)(36,47)(37,48)(44,46);(10,14)(15,16)(17,20,24,21)(18,26,29,44)(19,31,34,32)(22,36,23,37)(25,40,43,46)(27,38,35,28)(30,45,42,41)(33,47,39,48);(9,11,13,12)(10,15,14,16)(17,21,24,20)(18,27,29,35)(19,32,34,31)(22,36,23,37)(25,41,43,45)(26,38,44,28)(30,46,42,40)(33,47,39,48);(1,3,4,7)(2,6,5,8)(9,12,13,11)(10,15,14,16)(17,22,24,23)(18,28,29,38)(19,33,34,39)(20,36,21,37)(25,42,43,30)(26,27,44,35)(31,47,32,48)(40,41,46,45)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6392865);