#9206521

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

4075 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=i,d^2=gh,e^2=hi,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,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, 9113717);