#9062146

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

4166 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8969342);