#9023567

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

4196 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8930763);