#8291649

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

4064 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)(3,8)(4,6)(5,9)(7,11)(10,15)(12,13)(14,16)(17,18,22,20)(19,23,25,26)(21,24,29,28)(27,30,31,32)(33,34,40,44)(35,45,46,39)(36,42,37,41)(38,48,47,43);(1,3)(2,7)(4,8)(5,10)(6,11)(9,14)(12,15)(13,16)(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,32)(29,31)(33,35,40,46)(34,39,44,45)(36,38,37,47)(41,48,42,43);(5,13)(9,12)(10,16)(14,15)(19,25)(23,26)(27,31)(30,32)(33,36,40,37)(34,41,44,42)(35,38,46,47)(39,48,45,43);(1,4,6,2)(3,8,11,7)(5,12,13,9)(10,15,16,14)(17,20,22,18)(19,26,25,23)(21,24,29,28)(27,30,31,32)(33,37,40,36)(34,42,44,41)(35,38,46,47)(39,48,45,43);(1,3)(2,8)(4,7)(5,10)(6,11)(9,15)(12,14)(13,16)(17,21)(18,24)(19,27)(20,28)(22,29)(23,30)(25,31)(26,32)(33,38)(34,43)(35,37)(36,46)(39,41)(40,47)(42,45)(44,48);(1,5,6,13)(2,9,4,12)(3,10,11,16)(7,14,8,15)(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,32)(29,31)(33,39,40,45)(34,35,44,46)(36,43,37,48)(38,41,47,42)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=h,d^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8198845);