#9913360

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

3961 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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=h,d^2=h,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, 9820556);