#483153

order	512
nilpotency class	4
isoclinism	isoclinic class of #482080
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#102
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	C₂×#41	C₂²⋊C₄
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ C₄×#131 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

695 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=gi,e^2=hi,f^2=ghi,h^2=i,b^a=be,c^a=cg,e^a=eg,e^b=eh,f^b=fgh,h^b=hi,d^c=di

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 390349);