#9913202

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

3884 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=gh,e^2=g,f^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, 9820398);