#9236456

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

3954 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:

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

Polycyclic presentation:

2^9|a^2=gi,c^2=gi,d^2=h,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,e^d=egi,f^d=fgi,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, 9143652);