#9210376

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

3801 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,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, 9117572);