#8455926

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

4170 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=ghi,d^2=g,e^2=i,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, 8363122);