#9164022

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

4003 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9071218);