#8750054

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

4251 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)(3,9)(4,8)(5,6)(7,11)(10,16)(12,14)(13,15)(17,18)(19,22)(20,25)(21,24)(23,28)(26,30)(27,32)(29,31)(33,34,39,43)(35,40,38,42)(36,41,47,37)(44,46,45,48);(2,6)(7,9)(8,12)(15,16)(17,18)(19,22)(20,23)(21,24)(25,28)(26,30)(27,31)(29,32)(33,35,39,38)(34,40,43,42)(36,45,47,44)(37,46,41,48);(2,6)(7,9)(8,12)(15,16)(18,19)(23,25)(24,26)(31,32)(33,36,39,47)(34,41,43,37)(35,44,38,45)(40,46,42,48);(1,2)(3,7)(4,12)(5,6)(8,14)(9,11)(10,16)(13,15)(17,19,22,18)(20,25,28,23)(21,24,30,26)(27,31,29,32)(33,36)(34,37)(35,45)(38,44)(39,47)(40,46)(41,43)(42,48);(1,3,5,11)(2,7,6,9)(4,13,14,10)(8,16,12,15)(17,20)(18,23)(19,25)(21,29)(22,28)(24,32)(26,31)(27,30)(33,37)(34,36)(35,46)(38,48)(39,41)(40,45)(42,44)(43,47);(1,4,5,14)(2,8,6,12)(3,10,11,13)(7,15,9,16)(17,21,22,30)(18,24,19,26)(20,27,28,29)(23,31,25,32)(33,38,39,35)(34,42,43,40)(36,45,47,44)(37,46,41,48)

Polycyclic presentation:

2^9|a^2=ghi,b^2=ghi,c^2=ghi,d^2=gh,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8657250);