#8718604

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

4167 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=gi,d^2=hi,e^2=h,f^2=g,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, 8625800);