#8464209

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

4149 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=hi,d^2=h,e^2=gi,f^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=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, 8371405);