#8997162

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

4258 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=hi,d^2=ghi,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=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, 8904358);