#8201991

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

4117 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:

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

Polycyclic presentation:

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

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, 8109187);