#9040821

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

4145 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8948017);