#8827931

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

4117 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)(3,7,10,4)(5,8,14,16)(11,15,13,12)(17,18)(19,25)(20,28)(21,22)(23,27)(24,29)(26,31)(30,32)(33,34,37,41)(35,38,43,39)(36,44,46,40)(42,48,45,47);(1,3,6,10)(2,7,9,4)(5,13,14,11)(8,12,16,15)(18,21)(20,29)(23,25)(26,30)(33,35)(34,38)(36,45)(37,43)(39,41)(40,48)(42,46)(44,47);(3,10)(4,7)(11,13)(12,15)(17,19,22,27)(18,23,21,25)(20,26,29,30)(24,31,28,32)(33,35,37,43)(34,39,41,38)(36,42,46,45)(40,48,44,47);(1,4)(2,3)(5,12)(6,7)(8,11)(9,10)(13,16)(14,15)(17,19)(18,23)(20,30)(21,25)(22,27)(24,32)(26,29)(28,31)(33,35)(34,38)(36,42)(37,43)(39,41)(40,47)(44,48)(45,46);(1,2)(3,7)(4,10)(5,8)(6,9)(11,15)(12,13)(14,16)(17,20)(18,24)(19,26)(21,28)(22,29)(23,31)(25,32)(27,30)(33,34)(35,39)(36,40)(37,41)(38,43)(42,48)(44,46)(45,47);(1,5)(2,8)(3,11)(4,12)(6,14)(7,15)(9,16)(10,13)(17,21,22,18)(19,25,27,23)(20,28,29,24)(26,32,30,31)(33,36)(34,40)(35,42)(37,46)(38,47)(39,48)(41,44)(43,45)

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=h,f^2=g,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, 8735127);