#9923654

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

4022 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=hi,d^2=hi,e^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9830850);