#9912855

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

3871 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=gh,d^2=h,e^2=h,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, 9820051);