#9909752

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

4279 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=gi,d^2=gi,e^2=ghi,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,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, 9816948);