#10106948

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

4068 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)(3,4)(5,13)(6,7)(8,10)(9,15)(11,14)(12,16)(17,18,22,19)(20,23,28,25)(21,26,30,24)(27,32,29,31)(33,34,39,40)(35,36,37,41)(38,46,47,42)(43,48,45,44);(2,7)(4,8)(9,13)(12,14)(17,18,22,19)(20,25,28,23)(21,24,30,26)(27,32,29,31)(34,40)(35,37)(42,46)(43,45);(2,7)(3,10)(9,13)(11,16)(18,19)(23,25)(24,26)(31,32)(33,35,39,37)(34,41,40,36)(38,43,47,45)(42,48,46,44);(1,3,6,10)(2,8,7,4)(5,11,15,16)(9,14,13,12)(17,19,22,18)(20,23,28,25)(21,26,30,24)(27,31,29,32)(33,36,39,41)(34,35,40,37)(38,44,47,48)(42,43,46,45);(1,4,6,8)(2,3,7,10)(5,14,15,12)(9,16,13,11)(17,20)(18,23)(19,25)(21,29)(22,28)(24,32)(26,31)(27,30)(33,37,39,35)(34,41,40,36)(38,45,47,43)(42,48,46,44);(1,5,6,15)(2,9,7,13)(3,11,10,16)(4,12,8,14)(17,21)(18,24)(19,26)(20,27)(22,30)(23,31)(25,32)(28,29)(33,38,39,47)(34,42,40,46)(35,43,37,45)(36,44,41,48)

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=ghi,d^2=h,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10014144);