#9026727

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

4062 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:

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

Polycyclic presentation:

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

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, 8933923);