#8745813

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

4092 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:

(1,2,6,9)(3,4,11,7)(5,8,15,16)(10,12,13,14)(17,18,22,26)(19,23,29,32)(20,30,28,24)(21,25,27,31)(34,38)(39,44)(40,45)(46,48);(1,3,6,11)(2,7,9,4)(5,10,15,13)(8,14,16,12)(17,19,22,29)(18,23,26,32)(20,21,28,27)(24,31,30,25)(35,41)(39,44)(42,47)(46,48);(1,3,6,11)(2,7,9,4)(5,10,15,13)(8,14,16,12)(17,20,22,28)(18,24,26,30)(19,27,29,21)(23,25,32,31)(33,34)(35,39)(36,40)(37,38)(41,44)(42,46)(43,45)(47,48);(1,3)(2,4)(5,13)(6,11)(7,9)(8,14)(10,15)(12,16)(17,20)(18,24)(19,27)(21,29)(22,28)(23,25)(26,30)(31,32)(33,35,37,41)(34,39,38,44)(36,42,43,47)(40,46,45,48);(1,4)(2,3)(5,14)(6,7)(8,13)(9,11)(10,16)(12,15)(17,18)(19,23)(20,24)(21,31)(22,26)(25,27)(28,30)(29,32)(33,34)(35,39)(36,40)(37,38)(41,44)(42,46)(43,45)(47,48);(1,5)(2,8)(3,10)(4,12)(6,15)(7,14)(9,16)(11,13)(17,21,22,27)(18,25,26,31)(19,28,29,20)(23,24,32,30)(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=gi,c^2=gi,d^2=ghi,f^2=gh,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

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, 8653009);