#8127311

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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=gi,d^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^c=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8034507);