#461217

order	512
nilpotency class	3
isoclinism	isoclinic class of #454426
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

899 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=fi,d^2=e,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^c=de,e^c=ei,f^c=fi,g^c=gi,h^c=hi,h^d=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 368413);