#9002446

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	#2678	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4001 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)(3,7,10,5)(4,8,12,15)(11,13,16,14)(17,18,23,27)(19,24,22,32)(20,21,26,25)(28,30,29,31)(33,34)(35,38)(36,41)(37,42)(39,43)(40,46)(44,45)(47,48);(1,3)(2,7)(4,11)(5,9)(6,10)(8,13)(12,16)(14,15)(17,19,23,22)(18,24,27,32)(20,29,26,28)(21,31,25,30)(33,35,39,46)(34,40,43,38)(36,44,42,48)(37,45,41,47);(3,10)(4,12)(5,7)(8,15)(17,20,23,26)(18,25,27,21)(19,28,22,29)(24,31,32,30)(33,36,39,42)(34,41,43,37)(35,44,46,48)(38,45,40,47);(1,4)(2,8)(3,11)(5,14)(6,12)(7,13)(9,15)(10,16)(17,20)(18,21)(19,29)(22,28)(23,26)(24,31)(25,27)(30,32)(33,36)(34,37)(35,44)(38,47)(39,42)(40,45)(41,43)(46,48);(1,5,6,7)(2,3,9,10)(4,13,12,14)(8,16,15,11)(17,21)(18,26)(19,30)(20,27)(22,31)(23,25)(24,29)(28,32)(33,37)(34,42)(35,45)(36,43)(38,44)(39,41)(40,48)(46,47);(1,3)(2,7)(4,11)(5,9)(6,10)(8,13)(12,16)(14,15)(17,22,23,19)(18,24,27,32)(20,28,26,29)(21,31,25,30)(33,38)(34,35)(36,47)(37,44)(39,40)(41,48)(42,45)(43,46)

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=gh,e^2=hi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8909642);