#8898582

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

4170 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=gi,d^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 8805778);