#9290534

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

4190 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)(3,9)(4,8)(5,6)(7,11)(10,15)(12,13)(14,16)(17,18)(19,24)(20,26)(21,22)(23,25)(27,32)(28,29)(30,31)(33,34,38,40)(35,39,43,46)(36,37,44,45)(41,42,47,48);(2,6)(7,9)(8,13)(14,15)(17,19,23,20)(18,24,25,26)(21,30,27,28)(22,31,32,29)(33,35,38,43)(34,39,40,46)(36,41,44,47)(37,42,45,48);(4,12)(8,13)(10,16)(14,15)(18,25)(19,20)(21,27)(29,31)(34,40)(36,44)(39,46)(41,47);(1,3,5,11)(2,7,6,9)(4,10,12,16)(8,14,13,15)(17,20,23,19)(18,26,25,24)(21,30,27,28)(22,31,32,29)(33,36)(34,37)(35,41)(38,44)(39,42)(40,45)(43,47)(46,48);(1,4)(2,8)(3,10)(5,12)(6,13)(7,14)(9,15)(11,16)(17,21,23,27)(18,22,25,32)(19,28,20,30)(24,29,26,31)(33,35)(34,39)(36,41)(37,42)(38,43)(40,46)(44,47)(45,48);(1,2)(3,7)(4,8)(5,6)(9,11)(10,14)(12,13)(15,16)(17,22)(18,27)(19,29)(20,31)(21,25)(23,32)(24,30)(26,28)(33,37)(34,36)(35,42)(38,45)(39,41)(40,44)(43,48)(46,47)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,d^2=gi,e^2=i,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dh,e^c=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, 9197730);