#8889108

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

4043 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

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

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, 8796304);