#8581465

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

4292 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=gi,d^2=hi,e^2=ghi,f^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8488661);