#9913022

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

4425 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)(3,8)(4,9)(5,10)(6,7)(11,13)(12,16)(14,15)(17,18,22,23)(19,21,25,29)(20,24,27,30)(26,28,31,32)(33,34,39,41)(35,38,36,40)(37,42,46,48)(43,45,44,47);(2,7)(4,8)(10,15)(12,13)(20,27)(24,30)(26,31)(28,32)(33,35,39,36)(34,40,41,38)(37,44,46,43)(42,45,48,47);(18,23)(19,25)(24,30)(26,31)(34,41)(35,36)(42,48)(43,44);(1,3,6,9)(2,8,7,4)(5,11,14,16)(10,13,15,12)(17,19)(18,21)(20,26)(22,25)(23,29)(24,28)(27,31)(30,32)(33,36,39,35)(34,38,41,40)(37,44,46,43)(42,45,48,47);(1,4,6,8)(2,9,7,3)(5,13,14,12)(10,11,15,16)(17,20)(18,24)(19,26)(21,28)(22,27)(23,30)(25,31)(29,32)(33,37,39,46)(34,42,41,48)(35,43,36,44)(38,45,40,47);(1,5)(2,10)(3,11)(4,12)(6,14)(7,15)(8,13)(9,16)(17,21)(18,19)(20,28)(22,29)(23,25)(24,26)(27,32)(30,31)(33,38)(34,35)(36,41)(37,45)(39,40)(42,43)(44,48)(46,47)

Polycyclic presentation:

2^9|a^2=h,b^2=i,d^2=g,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, 9820218);