#8139474

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

4051 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)(3,6)(4,9)(5,8)(7,12)(10,13)(11,15)(14,16)(17,18,21,25)(19,23,26,22)(20,28,29,24)(27,30,32,31)(33,34)(35,43)(36,47)(37,39)(38,48)(40,46)(41,45)(42,44);(2,3)(7,9)(8,10)(14,15)(19,26)(22,23)(27,32)(30,31)(33,34,39,37)(35,40,46,43)(36,47,45,41)(38,42,44,48);(1,2,6,3)(4,7,12,9)(5,8,13,10)(11,14,16,15)(17,19)(18,22)(20,27)(21,26)(23,25)(24,30)(28,31)(29,32)(33,35)(34,40)(36,44)(37,43)(38,45)(39,46)(41,42)(47,48);(1,3,6,2)(4,9,12,7)(5,10,13,8)(11,15,16,14)(17,19)(18,23)(20,27)(21,26)(22,25)(24,31)(28,30)(29,32)(33,36,39,45)(34,41,37,47)(35,44,46,38)(40,42,43,48);(1,4,6,12)(2,7,3,9)(5,11,13,16)(8,14,10,15)(17,18)(19,22)(20,24)(21,25)(23,26)(27,30)(28,29)(31,32)(33,37,39,34)(35,43,46,40)(36,41,45,47)(38,48,44,42);(1,5,6,13)(2,8,3,10)(4,11,12,16)(7,14,9,15)(17,20,21,29)(18,24,25,28)(19,27,26,32)(22,30,23,31)(33,38,39,44)(34,42,37,48)(35,45,46,36)(40,41,43,47)

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=ghi,d^2=g,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,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, 8046670);