#6485723

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

1406 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)(19,29)(20,26)(21,27)(22,36)(23,28)(24,31)(25,33)(30,38)(32,40)(34,42)(35,43)(37,41)(39,45)(44,47)(46,48);(2,5)(6,8)(9,10)(11,16)(12,15)(13,14)(17,19,23,32)(18,24,28,41)(20,30,34,46)(21,31,22,37)(25,38,43,48)(26,39,42,47)(27,40,36,29)(33,44,35,45);(10,14)(15,16)(17,20,23,34)(18,25,28,43)(19,30,32,46)(21,33,22,35)(24,38,41,48)(26,27,42,36)(29,39,40,47)(31,44,37,45);(9,11,13,12)(10,15,14,16)(17,20)(18,26)(19,30)(21,35)(22,33)(23,34)(24,39)(25,36)(27,43)(28,42)(29,38)(31,45)(32,46)(37,44)(40,48)(41,47);(1,3,4,7)(2,6,5,8)(9,12,13,11)(10,15,14,16)(17,21,23,22)(18,27,28,36)(19,31,32,37)(20,33,34,35)(24,40,41,29)(25,42,43,26)(30,44,46,45)(38,47,48,39)

Polycyclic presentation:

2^9|b^2=i,c^2=i,d^2=h,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,d^c=di,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, 6392919);