#8873728

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

4106 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)(4,9,12,7)(5,10,13,8)(11,14,16,15)(17,18,22,24)(19,20,27,23)(21,29,31,25)(26,32,30,28)(33,34,38,41)(35,42,39,36)(37,40,46,48)(43,47,45,44);(1,2,6,3)(4,9,12,7)(5,10,13,8)(11,14,16,15)(17,19,22,27)(18,23,24,20)(21,30,31,26)(25,28,29,32)(33,35,38,39)(34,36,41,42)(37,45,46,43)(40,47,48,44);(2,3)(7,9)(8,10)(14,15)(18,24)(20,23)(25,29)(28,32)(33,34,38,41)(35,36,39,42)(37,40,46,48)(43,44,45,47);(1,3,6,2)(4,7,12,9)(5,10,13,8)(11,14,16,15)(17,20)(18,19)(21,28)(22,23)(24,27)(25,26)(29,30)(31,32)(33,34)(35,36)(37,40)(38,41)(39,42)(43,44)(45,47)(46,48);(1,4,6,12)(2,7,3,9)(5,11,13,16)(8,14,10,15)(17,19)(18,23)(20,24)(21,26)(22,27)(25,32)(28,29)(30,31)(33,36)(34,39)(35,41)(37,44)(38,42)(40,45)(43,48)(46,47);(1,5)(2,8)(3,10)(4,11)(6,13)(7,14)(9,15)(12,16)(17,21)(18,25)(19,26)(20,28)(22,31)(23,32)(24,29)(27,30)(33,37)(34,40)(35,43)(36,44)(38,46)(39,45)(41,48)(42,47)

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8780924);