#8718167

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

4213 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)(4,8,14,12)(10,16,13,15)(17,18,22,23)(19,20,26,25)(21,24,31,32)(27,28,29,30)(33,34,39,43)(35,40,44,47)(36,41,45,46)(37,38,42,48);(2,6)(7,9)(8,12)(15,16)(17,18,22,23)(19,25,26,20)(21,24,31,32)(27,30,29,28)(33,35,39,44)(34,40,43,47)(36,37,45,42)(38,46,48,41);(2,6)(7,9)(8,12)(15,16)(18,23)(19,26)(24,32)(27,29)(35,44)(36,45)(40,47)(41,46);(1,2)(3,7)(4,12)(5,6)(8,14)(9,11)(10,16)(13,15)(17,19)(18,20)(21,29)(22,26)(23,25)(24,30)(27,31)(28,32)(33,36,39,45)(34,41,43,46)(35,37,44,42)(38,47,48,40);(1,3,5,11)(2,7,6,9)(4,13,14,10)(8,16,12,15)(17,20)(18,19)(21,30)(22,25)(23,26)(24,29)(27,32)(28,31)(33,37)(34,38)(35,45)(36,44)(39,42)(40,46)(41,47)(43,48);(1,4,5,14)(2,8,6,12)(3,10,11,13)(7,15,9,16)(17,21,22,31)(18,24,23,32)(19,27,26,29)(20,28,25,30)(33,38)(34,42)(35,46)(36,40)(37,43)(39,48)(41,44)(45,47)

Polycyclic presentation:

2^9|a^2=gi,b^2=i,d^2=ghi,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, 8625363);