#10195798

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

4039 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:

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

Polycyclic presentation:

2^9|a^2=ghi,c^2=i,d^2=h,e^2=gh,f^2=ghi,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, 10102994);