#9210281

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3807 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)(3,8,11,12)(4,13,14,9)(5,10,15,16)(17,18)(19,26)(20,25)(21,24)(22,23)(27,32)(28,31)(29,30)(33,34,39,37)(35,40,45,46)(36,42,38,41)(43,48,44,47);(1,3)(2,8)(4,5)(6,12)(7,11)(9,16)(10,13)(14,15)(17,18,23,22)(19,24,21,26)(20,25,30,29)(27,31,28,32)(33,35)(34,40)(36,43)(37,46)(38,44)(39,45)(41,47)(42,48);(3,11)(4,14)(8,12)(9,13)(17,19,23,21)(18,24,22,26)(20,28,30,27)(25,32,29,31)(33,36,39,38)(34,41,37,42)(35,43,45,44)(40,47,46,48);(1,4,7,14)(2,9,6,13)(3,5,11,15)(8,16,12,10)(17,20,23,30)(18,25,22,29)(19,27,21,28)(24,31,26,32);(1,5)(2,10)(3,4)(6,16)(7,15)(8,13)(9,12)(11,14)(17,21,23,19)(18,24,22,26)(20,27,30,28)(25,32,29,31)(33,37,39,34)(35,40,45,46)(36,42,38,41)(43,47,44,48);(1,6,7,2)(3,12,11,8)(4,13,14,9)(5,10,15,16)(17,22,23,18)(19,26,21,24)(20,29,30,25)(27,32,28,31)(33,38,39,36)(34,42,37,41)(35,44,45,43)(40,48,46,47)

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=h,d^2=gi,e^2=h,f^2=hi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,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, 9117477);