#6488805

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470003
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1627 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)(11,14)(12,15)(13,16)(17,18,22,27,23,28,34,41)(19,29,32,40,33,44,37,24)(20,25,21,26,35,42,36,43)(30,45,47,48,46,39,31,38);(11,12)(14,15)(17,19,23,33)(18,24,28,40)(20,30,35,46)(21,31,36,47)(22,37,34,32)(25,38,42,48)(26,39,43,45)(27,44,41,29);(2,5)(6,8)(9,11,13,12)(10,14,16,15)(17,20)(18,25)(19,30)(21,22)(23,35)(24,38)(26,27)(28,42)(29,45)(31,37)(32,47)(33,46)(34,36)(39,44)(40,48)(41,43);(9,12,13,11)(10,15,16,14)(17,21)(18,26)(19,31)(20,34)(22,35)(23,36)(24,39)(25,41)(27,42)(28,43)(29,38)(30,32)(33,47)(37,46)(40,45)(44,48);(1,3,4,7)(2,6,5,8)(10,16)(14,15)(20,35)(21,36)(25,42)(26,43)(30,46)(31,47)(38,48)(39,45)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=h,d^2=h,e^2=g,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=dh,f^b=fi,d^c=di,e^c=ei,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, 6396001);