#477609

order	512
nilpotency class	3
isoclinism	isoclinic class of #476092
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

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

879 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)(3,7,8,6)(11,15)(13,16)(17,18,22,28)(19,30,33,25)(20,26,21,36)(23,29,24,40)(27,35,45,38)(31,46,47,41)(32,48,37,44)(34,43,39,42);(2,5)(6,7)(9,10,12,14)(11,13,15,16)(17,19,22,33)(18,25,28,30)(20,32,21,37)(23,39,24,34)(26,44,36,48)(27,46,45,41)(29,43,40,42)(31,38,47,35);(1,3,4,8)(2,6,5,7)(9,11,12,15)(10,13,14,16)(17,20)(18,26)(19,31)(21,22)(23,38)(24,35)(25,41)(27,29)(28,36)(30,46)(32,39)(33,47)(34,37)(40,45)(42,48)(43,44);(3,8)(6,7)(10,14)(13,16)(17,21,22,20)(18,27,28,45)(19,32,33,37)(23,38,24,35)(25,41,30,46)(26,40,36,29)(31,34,47,39)(42,44,43,48)

Polycyclic presentation:

2^9|a^2=e,b^2=ef,c^2=f,d^2=i,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, 384805);