#9936385

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

4110 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)(4,8)(5,6)(9,13)(10,15)(11,16)(12,14)(17,18)(19,24)(20,25)(21,29)(22,23)(26,31)(27,28)(30,32)(33,34,39,40)(35,38,36,43)(37,41,46,48)(42,47,44,45);(2,6)(3,9)(8,14)(10,11)(18,23)(19,25)(26,29)(27,32)(33,35)(34,38)(36,39)(37,42)(40,43)(41,47)(44,46)(45,48);(18,23)(19,25)(26,29)(27,32)(34,40)(35,36)(41,48)(42,44);(1,2)(3,7)(4,8)(5,6)(9,13)(10,15)(11,16)(12,14)(17,19,22,25)(18,24,23,20)(21,27,31,32)(26,30,29,28)(33,36,39,35)(34,38,40,43)(37,44,46,42)(41,47,48,45);(1,3,5,9)(2,7,6,13)(4,11,12,10)(8,16,14,15)(17,20,22,24)(18,25,23,19)(21,30,31,28)(26,27,29,32)(33,37)(34,41)(35,42)(36,44)(38,47)(39,46)(40,48)(43,45);(1,4,5,12)(2,8,6,14)(3,10,9,11)(7,15,13,16)(17,21,22,31)(18,26,23,29)(19,27,25,32)(20,28,24,30)(33,38)(34,35)(36,40)(37,45)(39,43)(41,44)(42,48)(46,47)

Polycyclic presentation:

2^9|a^2=i,d^2=h,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, 9843581);