#483138

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

677 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)(17,18,22,27,25,32,39,42,23,28,21,38,24,19,29,31)(20,33,36,47,40,30,44,46,34,26,35,41,37,45,43,48);(1,3,5,11)(2,6,7,9)(4,12,13,10)(8,16,15,14)(17,19,23,32)(18,25,28,24)(20,30,34,45)(21,38,22,27)(26,40,33,37)(29,42,39,31)(35,47,36,41)(43,48,44,46);(2,7)(6,9)(8,15)(14,16)(17,20)(18,26)(19,30)(21,35)(22,36)(23,34)(24,37)(25,40)(27,41)(28,33)(29,43)(31,46)(32,45)(38,47)(39,44)(42,48);(20,34)(26,33)(30,45)(35,36)(37,40)(41,47)(43,44)(46,48)

Polycyclic presentation:

2^9|a^2=f,b^2=g,e^2=hi,f^2=gh,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, 390334);