#9167737

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

4220 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)(3,8)(4,9)(5,14)(6,7)(10,16)(11,15)(12,13)(17,18)(19,22)(20,24)(21,25)(23,26)(27,30)(28,29)(31,32)(35,42)(37,45)(39,40)(41,44);(2,7)(4,13)(8,15)(10,14)(17,19)(18,22)(20,27)(21,26)(23,25)(24,30)(28,31)(29,32)(33,34)(35,40)(36,38)(37,41)(39,42)(43,47)(44,45)(46,48);(2,7)(3,11)(9,12)(10,14)(19,26)(22,23)(27,32)(30,31)(33,35,38,42)(34,39,36,40)(37,43,45,48)(41,46,44,47);(1,3,6,11)(2,8,7,15)(4,14,13,10)(5,12,16,9)(17,19)(18,23)(20,27)(21,26)(22,25)(24,31)(28,30)(29,32)(33,35)(34,40)(36,39)(37,43)(38,42)(41,47)(44,46)(45,48);(1,4,6,13)(2,9,7,12)(3,10,11,14)(5,8,16,15)(17,18)(19,22)(20,28)(21,25)(23,26)(24,29)(27,31)(30,32)(33,36,38,34)(35,40,42,39)(37,41,45,44)(43,46,48,47);(1,5,6,16)(2,10,7,14)(3,12,11,9)(4,15,13,8)(17,20)(18,24)(19,27)(21,29)(22,30)(23,31)(25,28)(26,32)(33,37)(34,41)(35,43)(36,44)(38,45)(39,46)(40,47)(42,48)

Polycyclic presentation:

2^9|c^2=gh,d^2=g,e^2=h,f^2=g,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, 9074933);