#8806902

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

4214 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)(3,6)(4,11)(5,7)(8,13)(9,15)(10,16)(12,14)(17,18)(19,26)(20,23)(21,24)(22,28)(25,31)(27,29)(30,32)(33,34,38,39)(35,40,43,37)(36,41,46,47)(42,48,44,45);(1,3,5,9)(2,6,7,15)(4,12,13,10)(8,16,11,14)(17,19,23,24)(18,21,20,26)(22,30,31,27)(25,32,28,29)(34,39)(37,40)(41,47)(45,48);(2,7)(3,9)(8,11)(10,12)(18,20)(21,26)(25,28)(29,32)(33,35,38,43)(34,40,39,37)(36,44,46,42)(41,45,47,48);(1,2)(3,6)(4,8)(5,7)(9,15)(10,14)(11,13)(12,16)(17,20,23,18)(19,26,24,21)(22,28,31,25)(27,32,30,29)(33,35)(34,37)(36,42)(38,43)(39,40)(41,45)(44,46)(47,48);(1,3)(2,6)(4,10)(5,9)(7,15)(8,14)(11,16)(12,13)(17,21,23,26)(18,24,20,19)(22,29,31,32)(25,30,28,27)(33,36)(34,41)(35,42)(37,45)(38,46)(39,47)(40,48)(43,44);(1,4,5,13)(2,8,7,11)(3,10,9,12)(6,14,15,16)(17,22,23,31)(18,25,20,28)(19,27,24,30)(21,29,26,32)(33,37)(34,35)(36,45)(38,40)(39,43)(41,42)(44,47)(46,48)

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=ghi,d^2=hi,e^2=hi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,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, 8714098);