#8436676

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

4024 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)(17,18,22,24)(19,25,27,21)(20,23,29,30)(26,31,32,28)(33,34)(35,40)(36,47)(37,41)(38,42)(39,43)(44,45)(46,48);(1,3,6,5)(2,7,9,10)(4,11,14,13)(8,15,12,16)(17,19,22,27)(18,21,24,25)(20,26,29,32)(23,28,30,31)(33,35,39,46)(34,40,43,48)(36,37,42,44)(38,45,47,41);(17,18)(19,21)(20,23)(22,24)(25,27)(26,28)(29,30)(31,32)(33,36,39,42)(34,38,43,47)(35,44,46,37)(40,41,48,45);(1,2)(3,7)(4,8)(5,10)(6,9)(11,15)(12,14)(13,16)(17,18)(19,25)(20,23)(21,27)(22,24)(26,31)(28,32)(29,30)(33,35)(34,40)(36,44)(37,42)(38,41)(39,46)(43,48)(45,47);(1,4)(2,8)(3,11)(5,13)(6,14)(7,15)(9,12)(10,16)(17,20,22,29)(18,23,24,30)(19,26,27,32)(21,28,25,31)(33,37,39,44)(34,41,43,45)(35,42,46,36)(38,48,47,40);(1,5,6,3)(2,7,9,10)(4,13,14,11)(8,15,12,16)(17,21,22,25)(18,19,24,27)(20,28,29,31)(23,26,30,32)(33,38)(34,42)(35,45)(36,43)(37,48)(39,47)(40,44)(41,46)

Polycyclic presentation:

2^9|a^2=g,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, 8343872);