#9369119

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
prime radical	#52372	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4069 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=gh,d^2=h,e^2=g,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=dghi,e^d=egi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9276315);