#9730615

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

3997 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=hi,d^2=g,e^2=gi,f^2=ghi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,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, 9637811);