#8505436

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

4151 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)(3,7)(4,13)(5,8)(6,9)(10,16)(11,14)(12,15)(17,18,23,20)(19,24,29,27)(21,30,32,25)(22,26,28,31)(33,34,39,40)(35,41,44,46)(36,45,43,42)(37,47,48,38);(1,3,6,12)(2,7,9,15)(4,11,5,10)(8,16,13,14)(17,19)(18,24)(20,27)(21,22)(23,29)(25,31)(26,30)(28,32)(34,40)(37,48)(41,46)(42,45);(1,4,6,5)(2,8,9,13)(3,10,12,11)(7,14,15,16)(17,18,23,20)(19,27,29,24)(21,25,32,30)(22,26,28,31)(33,35,39,44)(34,41,40,46)(36,38,43,47)(37,42,48,45);(1,5,6,4)(2,8,9,13)(3,11,12,10)(7,14,15,16)(17,20,23,18)(19,27,29,24)(21,30,32,25)(22,26,28,31)(33,36)(34,42)(35,38)(37,46)(39,43)(40,45)(41,48)(44,47);(1,3)(2,7)(4,10)(5,11)(6,12)(8,14)(9,15)(13,16)(17,21)(18,25)(19,28)(20,30)(22,29)(23,32)(24,26)(27,31)(33,37)(34,38)(35,42)(36,46)(39,48)(40,47)(41,43)(44,45);(1,2)(3,7)(4,8)(5,13)(6,9)(10,14)(11,16)(12,15)(17,22,23,28)(18,26,20,31)(19,21,29,32)(24,30,27,25)(33,38)(34,37)(35,43)(36,44)(39,47)(40,48)(41,42)(45,46)

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=gi,d^2=i,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, 8412632);