#9018073

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

4282 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:

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

Polycyclic presentation:

2^9|a^2=gh,c^2=h,f^2=i,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, 8925269);