#9937771

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

4093 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)(4,10)(5,13)(6,8)(9,12)(11,16)(14,15)(18,22)(20,28)(23,25)(26,32)(33,34,38,41)(35,39,43,36)(37,45,47,40)(42,44,48,46);(1,3,6,12)(2,7,8,9)(4,13,15,11)(5,14,16,10)(17,18)(19,25)(20,24)(21,22)(23,27)(26,31)(28,29)(30,32)(33,34)(35,36)(37,40)(38,41)(39,43)(42,44)(45,47)(46,48);(2,8)(3,12)(10,14)(11,13)(17,18,21,22)(19,23,27,25)(20,24,28,29)(26,30,32,31)(33,35,38,43)(34,36,41,39)(37,42,47,48)(40,44,45,46);(1,3)(2,9)(4,13)(5,14)(6,12)(7,8)(10,16)(11,15)(17,18)(19,23)(20,24)(21,22)(25,27)(26,30)(28,29)(31,32)(33,35)(34,39)(36,41)(37,42)(38,43)(40,46)(44,45)(47,48);(1,4)(2,10)(3,13)(5,7)(6,15)(8,14)(9,16)(11,12)(17,19,21,27)(18,23,22,25)(20,26,28,32)(24,30,29,31)(33,36,38,39)(34,35,41,43)(37,46,47,44)(40,48,45,42);(1,5,6,16)(2,11,8,13)(3,14,12,10)(4,9,15,7)(17,20,21,28)(18,24,22,29)(19,26,27,32)(23,30,25,31)(33,37,38,47)(34,40,41,45)(35,42,43,48)(36,44,39,46)

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=g,e^2=g,f^2=h,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, 9844967);