#9712319

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
prime radical	#43092	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4017 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:

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

Polycyclic presentation:

2^9|b^2=hi,c^2=gi,d^2=gi,e^2=g,f^2=i,b^a=bg,c^a=ch,d^a=dg,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, 9619515);