#9058281

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

4176 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:

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

Polycyclic presentation:

2^9|b^2=h,c^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,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, 8965477);