#8664729

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

4641 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=hi,d^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8571925);