#9913340

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

3896 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)(3,9)(4,8)(5,7)(6,11)(10,16)(12,14)(13,15)(17,18)(19,21)(20,25)(22,26)(23,24)(27,29)(28,30)(31,32)(35,41)(38,45)(42,47)(46,48);(1,3)(2,6)(4,10)(5,11)(7,9)(8,14)(12,13)(15,16)(18,24)(21,25)(26,32)(29,30)(33,34,37,40)(35,38,41,45)(36,39,44,43)(42,46,47,48);(2,7)(6,9)(8,15)(14,16)(17,19,23,20)(18,21,24,25)(22,27,31,28)(26,29,32,30)(33,34)(35,38)(36,39)(37,40)(41,45)(42,46)(43,44)(47,48);(1,2)(3,6)(4,8)(5,7)(9,11)(10,14)(12,16)(13,15)(17,20,23,19)(18,25,24,21)(22,28,31,27)(26,30,32,29)(33,35)(34,38)(36,42)(37,41)(39,46)(40,45)(43,48)(44,47);(1,4,5,13)(2,8,7,15)(3,10,11,12)(6,14,9,16)(17,21,23,25)(18,19,24,20)(22,30,31,29)(26,28,32,27)(33,34)(35,38)(36,43)(37,40)(39,44)(41,45)(42,48)(46,47);(1,3)(2,6)(4,12)(5,11)(7,9)(8,16)(10,13)(14,15)(17,22)(18,26)(19,27)(20,28)(21,29)(23,31)(24,32)(25,30)(33,36)(34,39)(35,42)(37,44)(38,46)(40,43)(41,47)(45,48)

Polycyclic presentation:

2^9|b^2=i,c^2=gh,d^2=gh,e^2=g,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, 9820536);