#9054478

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

4308 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)(3,4,8,11)(5,9,13,14)(10,12,15,16)(17,18,23,27)(19,24,28,22)(20,21,26,25)(29,31,32,30)(33,34)(35,39)(36,44)(37,41)(38,43)(40,45)(42,46)(47,48);(2,7)(4,11)(9,14)(12,16)(17,19,23,28)(18,24,27,22)(20,29,26,32)(21,31,25,30)(33,35)(34,38)(36,42)(37,43)(39,41)(40,46)(44,47)(45,48);(3,8)(4,11)(5,13)(9,14)(19,28)(20,26)(21,25)(22,24)(33,34,37,41)(35,38,43,39)(36,44,45,40)(42,47,48,46);(1,3)(2,4)(5,10)(6,8)(7,11)(9,12)(13,15)(14,16)(17,20)(18,25)(19,29)(21,27)(22,31)(23,26)(24,30)(28,32)(33,34)(35,38)(36,40)(37,41)(39,43)(42,46)(44,45)(47,48);(1,4)(2,8)(3,7)(5,12)(6,11)(9,15)(10,14)(13,16)(17,21,23,25)(18,26,27,20)(19,30,28,31)(22,32,24,29)(33,35)(34,39)(36,42)(37,43)(38,41)(40,47)(44,46)(45,48);(1,5,6,13)(2,9,7,14)(3,10,8,15)(4,12,11,16)(17,22)(18,19)(20,31)(21,32)(23,24)(25,29)(26,30)(27,28)(33,36,37,45)(34,40,41,44)(35,42,43,48)(38,46,39,47)

Polycyclic presentation:

2^9|a^2=ghi,b^2=i,c^2=h,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8961674);