#8436724

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

4026 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=i,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8343920);