#8463300

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

4017 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=g,e^2=i,f^2=hi,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, 8370496);