#9912895

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

3948 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)(3,7)(4,11)(5,6)(8,12)(9,13)(10,16)(14,15)(17,18)(19,23)(20,24)(21,22)(25,30)(26,29)(27,28)(31,32)(33,34,39,41)(35,45,46,40)(36,37,43,42)(38,48,47,44);(3,9)(7,13)(10,15)(14,16)(18,22)(23,29)(24,27)(30,31)(33,35)(34,40)(36,47)(37,48)(38,43)(39,46)(41,45)(42,44);(2,6)(7,13)(8,11)(14,16)(34,41)(35,46)(36,43)(44,48);(1,2)(3,7)(4,8)(5,6)(9,13)(10,14)(11,12)(15,16)(17,19,21,26)(18,23,22,29)(20,25,28,32)(24,30,27,31)(33,36,39,43)(34,42,41,37)(35,47,46,38)(40,44,45,48);(1,3,5,9)(2,7,6,13)(4,10,12,15)(8,14,11,16)(17,18)(19,23)(20,27)(21,22)(24,28)(25,31)(26,29)(30,32)(33,37,39,42)(34,43,41,36)(35,44,46,48)(38,40,47,45);(1,4,5,12)(2,8,6,11)(3,10,9,15)(7,14,13,16)(17,20)(18,24)(19,25)(21,28)(22,27)(23,30)(26,32)(29,31)(33,38,39,47)(34,44,41,48)(35,43,46,36)(37,40,42,45)

Polycyclic presentation:

2^9|a^2=i,d^2=g,e^2=h,f^2=h,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, 9820091);