#8534476

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4268 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=hi,d^2=h,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8441672);