#8942246

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

4048 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,d^2=h,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 8849442);