#8399491

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

4119 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=h,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8306687);