#477668

order	512
nilpotency class	3
isoclinism	isoclinic class of #476092
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

766 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 384864);