#483229

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

694 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,d^2=g,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, 390425);