#9783164

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

4037 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=g,d^2=h,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,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, 9690360);