#9017605

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

4170 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)(3,9)(4,12)(5,6)(7,10)(8,13)(11,14)(15,16)(17,18,22,24)(19,23,27,32)(20,25,29,28)(21,30,31,26)(33,34,38,40)(35,41,43,37)(36,39,46,44)(42,47,48,45);(2,6)(7,9)(8,12)(14,15)(17,19,22,27)(18,23,24,32)(20,26,29,30)(21,28,31,25)(33,35)(34,37)(36,42)(38,43)(39,45)(40,41)(44,47)(46,48);(2,6)(3,10)(4,13)(14,15)(18,24)(19,27)(20,29)(21,31)(33,34,38,40)(35,37,43,41)(36,44,46,39)(42,47,48,45);(1,3,5,10)(2,7,6,9)(4,11,13,16)(8,14,12,15)(17,20)(18,25)(19,26)(21,23)(22,29)(24,28)(27,30)(31,32)(33,34)(35,37)(36,44)(38,40)(39,46)(41,43)(42,47)(45,48);(1,2)(3,9)(4,8)(5,6)(7,10)(11,15)(12,13)(14,16)(17,19)(18,23)(20,30)(21,28)(22,27)(24,32)(25,31)(26,29)(33,36,38,46)(34,39,40,44)(35,42,43,48)(37,45,41,47);(1,4,5,13)(2,8,6,12)(3,11,10,16)(7,14,9,15)(17,21,22,31)(18,26,24,30)(19,28,27,25)(20,23,29,32)(33,37)(34,35)(36,45)(38,41)(39,42)(40,43)(44,48)(46,47)

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=ghi,d^2=hi,e^2=ghi,f^2=h,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, 8924801);