#483202

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	#3058	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

706 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

Polycyclic presentation:

2^9|a^2=f,d^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, 390398);