#9935889

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

4138 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=g,e^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, 9843085);