#8718321

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

4107 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=g,d^2=i,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,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, 8625517);