#8505482

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

4185 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)(3,11)(4,5)(6,8)(7,10)(9,15)(12,14)(13,16)(17,18,23,20)(19,24,29,27)(21,30,32,25)(22,26,28,31)(33,34,39,43)(35,44,46,40)(36,37,45,41)(38,42,48,47);(1,3,7,6)(2,8,10,11)(4,12,15,13)(5,16,9,14)(17,19)(18,24)(20,27)(21,22)(23,29)(25,31)(26,30)(28,32)(33,35,39,46)(34,40,43,44)(36,38,45,48)(37,47,41,42);(1,4)(2,9)(3,12)(5,10)(6,13)(7,15)(8,14)(11,16)(17,18,23,20)(19,27,29,24)(21,25,32,30)(22,26,28,31)(33,34,39,43)(35,40,46,44)(36,41,45,37)(38,42,48,47);(1,4)(2,9)(3,13)(5,10)(6,12)(7,15)(8,16)(11,14)(17,20,23,18)(19,27,29,24)(21,30,32,25)(22,26,28,31)(33,36)(34,41)(35,38)(37,43)(39,45)(40,42)(44,47)(46,48);(1,5,7,9)(2,4,10,15)(3,14,6,16)(8,13,11,12)(17,21)(18,25)(19,28)(20,30)(22,29)(23,32)(24,26)(27,31)(33,37,39,41)(34,36,43,45)(35,42,46,47)(38,40,48,44);(1,6,7,3)(2,8,10,11)(4,13,15,12)(5,16,9,14)(17,22,23,28)(18,26,20,31)(19,21,29,32)(24,30,27,25)(33,38)(34,42)(35,45)(36,46)(37,40)(39,48)(41,44)(43,47)

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=ghi,d^2=i,e^2=g,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=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, 8412678);