#8720743

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	C₂×#120	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3961 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8627939);