#9923656

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

3962 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)(3,8)(4,13)(5,10)(6,14)(7,9)(11,15)(12,16)(17,18)(19,25)(20,29)(21,23)(22,26)(24,27)(28,30)(31,32)(33,34,38,39)(35,36,40,43)(37,41,46,47)(42,44,48,45);(1,3,7,11)(2,8,9,15)(4,12,14,5)(6,10,13,16)(17,19,24,21)(18,25,27,23)(20,28,22,31)(26,32,29,30)(34,39)(36,43)(41,47)(44,45);(2,9)(3,11)(4,14)(10,16)(17,20,24,22)(18,26,27,29)(19,28,21,31)(23,30,25,32)(33,35,38,40)(34,36,39,43)(37,42,46,48)(41,44,47,45);(1,4)(2,6)(3,12)(5,11)(7,14)(8,10)(9,13)(15,16)(17,21,24,19)(18,23,27,25)(20,28,22,31)(26,32,29,30);(1,5,7,12)(2,10,9,16)(3,4,11,14)(6,8,13,15)(17,22,24,20)(18,26,27,29)(19,28,21,31)(23,32,25,30)(33,36)(34,40)(35,39)(37,45)(38,43)(41,42)(44,46)(47,48);(1,6)(2,4)(3,10)(5,15)(7,13)(8,12)(9,14)(11,16)(17,23)(18,19)(20,30)(21,27)(22,32)(24,25)(26,28)(29,31)(33,37)(34,41)(35,42)(36,44)(38,46)(39,47)(40,48)(43,45)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=ghi,d^2=hi,e^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9830852);