#9443213

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

4192 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)(3,7)(4,11)(5,6)(8,12)(9,13)(10,16)(14,15)(17,18,21,22)(19,23,27,25)(20,28,29,24)(26,31,32,30)(33,34)(35,42)(36,40)(37,45)(38,39)(41,47)(43,44)(46,48);(3,9)(7,13)(10,15)(14,16)(17,18)(19,25)(20,24)(21,22)(23,27)(26,31)(28,29)(30,32)(34,39)(36,42)(41,45)(44,48);(2,6)(7,13)(8,11)(14,16)(18,22)(23,25)(24,28)(30,31)(33,35,38,40)(34,36,39,42)(37,46,47,43)(41,48,45,44);(1,2)(3,7)(4,8)(5,6)(9,13)(10,14)(11,12)(15,16)(17,18)(19,23)(20,24)(21,22)(25,27)(26,30)(28,29)(31,32)(33,35)(34,36)(37,43)(38,40)(39,42)(41,44)(45,48)(46,47);(1,3)(2,7)(4,10)(5,9)(6,13)(8,14)(11,16)(12,15)(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,31)(29,32)(33,36)(34,40)(35,39)(37,44)(38,42)(41,46)(43,45)(47,48);(1,4)(2,8)(3,10)(5,12)(6,11)(7,14)(9,15)(13,16)(17,20,21,29)(18,24,22,28)(19,26,27,32)(23,30,25,31)(33,37,38,47)(34,41,39,45)(35,43,40,46)(36,44,42,48)

Polycyclic presentation:

2^9|a^2=i,c^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,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, 9350409);