#504188

order	512
nilpotency class	4
isoclinism	isoclinic class of #499712
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂⁴	C₂×D₈
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊C₄ ◁ C₈×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

528 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=fi,b^2=ef,c^2=fi,d^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gh,h^b=hi,g^c=gh,h^c=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 411384);