#9939589

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

4075 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)(3,7)(4,9)(5,10)(6,8)(11,12)(13,15)(14,16)(17,18)(19,28)(20,25)(21,26)(22,24)(23,27)(29,30)(31,32)(33,34)(35,38)(36,43)(37,40)(39,44)(41,47)(42,46)(45,48);(1,3,6,10)(2,7,8,5)(4,11,13,16)(9,12,15,14)(17,19,23,22)(18,24,27,28)(20,31,26,29)(21,32,25,30)(33,35,37,42)(34,38,40,46)(36,41,44,48)(39,45,43,47);(2,8)(3,10)(9,15)(11,16)(33,34,37,40)(35,38,42,46)(36,39,44,43)(41,45,48,47);(1,3)(2,5)(4,11)(6,10)(7,8)(9,14)(12,15)(13,16)(17,20,23,26)(18,25,27,21)(19,29,22,31)(24,32,28,30)(33,34)(35,38)(36,39)(37,40)(41,45)(42,46)(43,44)(47,48);(1,4)(2,9)(3,11)(5,14)(6,13)(7,12)(8,15)(10,16)(17,21,23,25)(18,26,27,20)(19,30,22,32)(24,31,28,29)(33,35)(34,38)(36,41)(37,42)(39,45)(40,46)(43,47)(44,48);(1,5)(2,3)(4,12)(6,7)(8,10)(9,16)(11,15)(13,14)(17,22,23,19)(18,24,27,28)(20,31,26,29)(21,30,25,32)(33,36)(34,39)(35,41)(37,44)(38,45)(40,43)(42,48)(46,47)

Polycyclic presentation:

2^9|b^2=gh,c^2=hi,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,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, 9846785);