#9909802

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
prime radical	#54866	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4169 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=i,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, 9816998);