#482333

order	512
nilpotency class	4
isoclinism	isoclinic class of #482080
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#102
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#60707	C₂
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

765 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)(3,9,11,6)(4,8)(5,7)(10,15,16,13)(12,14)(17,18)(19,29)(20,26)(21,35)(22,39)(23,28)(24,31)(25,32)(27,37)(30,45)(33,44)(34,46)(36,41)(38,40)(42,48)(43,47);(1,3)(2,6)(4,10)(5,11)(7,9)(8,13)(12,16)(14,15)(17,19)(18,24)(20,30)(21,36)(22,33)(23,32)(25,40)(26,41)(27,44)(28,43)(29,35)(31,37)(34,47)(38,48)(39,42)(45,46);(2,7)(6,9)(8,14)(13,15)(17,19)(18,25)(20,33)(21,31)(22,30)(23,32)(24,40)(26,45)(27,42)(28,29)(34,48)(35,43)(36,37)(38,47)(39,44)(41,46);(1,4,5,12)(2,8,7,14)(3,10,11,16)(6,13,9,15)(17,20,23,22)(18,26,28,39)(19,30,32,33)(21,34,38,37)(24,41,43,42)(25,44,29,45)(27,35,46,40)(31,36,47,48)

Polycyclic presentation:

2^9|a^2=f,d^2=gh,e^2=hi,g^2=i,h^2=i,b^a=be,c^a=cgh,e^a=eg,g^a=gi,h^a=hi,e^b=eh,f^b=fghi,g^b=gi,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, 389529);