#8691789

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

4244 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,d^2=g,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8598985);