#9506860

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

4149 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)(3,9,10,7)(5,12,13,8)(11,14,16,15)(17,18,23,25)(19,22,20,24)(21,26,30,32)(27,29,28,31)(33,34,38,40)(35,39,43,46)(36,37,44,45)(41,42,47,48);(2,4)(7,9)(8,12)(14,15)(17,19,23,20)(18,24,25,22)(21,28,30,27)(26,29,32,31)(33,35,38,43)(34,39,40,46)(36,41,44,47)(37,42,45,48);(2,4)(3,10)(5,13)(14,15)(18,25)(19,20)(26,32)(27,28)(34,40)(36,44)(39,46)(41,47);(1,3)(2,7)(4,9)(5,11)(6,10)(8,14)(12,15)(13,16)(17,20,23,19)(18,22,25,24)(21,28,30,27)(26,29,32,31)(33,36)(34,37)(35,41)(38,44)(39,42)(40,45)(43,47)(46,48);(1,4,6,2)(3,7,10,9)(5,12,13,8)(11,14,16,15)(17,21,23,30)(18,26,25,32)(19,27,20,28)(22,29,24,31)(33,35)(34,39)(36,41)(37,42)(38,43)(40,46)(44,47)(45,48);(1,5)(2,8)(3,11)(4,12)(6,13)(7,14)(9,15)(10,16)(17,22)(18,19)(20,25)(21,29)(23,24)(26,27)(28,32)(30,31)(33,37)(34,36)(35,42)(38,45)(39,41)(40,44)(43,48)(46,47)

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,d^2=i,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9414056);