#8605703

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

4170 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8512899);