#8139609

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	#46031	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4219 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:

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

Polycyclic presentation:

2^9|c^2=h,d^2=gi,e^2=g,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, 8046805);