#8534811

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

4365 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)(3,11)(4,13)(5,10)(6,7)(8,12)(9,14)(15,16)(18,23)(19,25)(24,30)(26,31)(33,34)(35,39)(36,43)(37,40)(38,41)(42,47)(44,48)(45,46);(2,7)(3,12)(9,13)(10,16)(17,18,22,23)(19,21,25,29)(20,24,27,30)(26,28,31,32)(33,35,38,43)(34,39,41,36)(37,42,46,48)(40,47,45,44);(2,7)(4,14)(8,11)(10,16)(20,27)(24,30)(26,31)(28,32)(33,36,38,39)(34,35,41,43)(37,44,46,47)(40,42,45,48);(1,3)(2,8)(4,10)(5,9)(6,12)(7,11)(13,15)(14,16)(17,19)(18,21)(20,26)(22,25)(23,29)(24,28)(27,31)(30,32)(33,34)(35,36)(37,45)(38,41)(39,43)(40,46)(42,47)(44,48);(1,4,6,14)(2,9,7,13)(3,10,12,16)(5,11,15,8)(17,20)(18,24)(19,26)(21,28)(22,27)(23,30)(25,31)(29,32)(33,36)(34,35)(37,44)(38,39)(40,42)(41,43)(45,48)(46,47);(1,5)(2,10)(3,13)(4,11)(6,15)(7,16)(8,14)(9,12)(17,21)(18,25)(19,23)(20,28)(22,29)(24,31)(26,30)(27,32)(33,37)(34,40)(35,42)(36,44)(38,46)(39,47)(41,45)(43,48)

Polycyclic presentation:

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

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, 8442007);