#499576

order	512
nilpotency class	4
isoclinism	isoclinic class of #13304
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂³	C₂²×D₈
prime radical	C₂⁴	C₂×D₈
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

536 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)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18)(19,24)(20,23)(21,25)(22,26)(27,39)(28,35)(29,40)(30,36)(31,37)(32,43)(33,42)(34,44)(38,46)(41,45)(47,48);(2,5)(6,8)(9,11,12,15)(10,13,14,16)(18,23)(19,27)(21,32)(22,31)(24,38)(25,41)(28,46)(29,47)(30,45)(33,34)(35,39)(36,43)(40,42)(44,48);(10,14)(13,16)(17,19,20,28)(18,24,23,35)(21,33,30,47)(22,34,31,29)(25,42,36,48)(26,44,37,40)(27,45,46,32)(38,43,39,41);(1,3,4,7)(2,6,5,8)

Polycyclic presentation:

2^9|b^2=f,c^2=i,d^2=ei,g^2=hi,h^2=i,b^a=be,c^a=cf,c^b=cg,g^b=gh,h^b=hi,g^c=gh,h^c=hi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406772);