#8702691

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

4598 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)(11,18)(12,20)(13,15)(14,17)(16,21)(19,23)(22,24)(25,26,31,34)(27,35,37,30)(28,38,40,32)(29,39,36,33);(1,2)(3,6)(4,7)(5,8)(9,11,14,13)(10,15,17,18)(12,19,21,22)(16,23,20,24)(25,27)(26,30)(28,29)(31,37)(32,39)(33,38)(34,35)(36,40);(1,3)(2,5)(4,8)(6,7)(12,21)(16,20)(19,22)(23,24)(25,28,31,40)(26,32,34,38)(27,29,37,36)(30,39,35,33);(3,8)(5,6)(9,12,14,21)(10,16,17,20)(11,19,13,22)(15,23,18,24)(25,28)(26,32)(27,36)(29,37)(30,33)(31,40)(34,38)(35,39);(1,2)(3,5)(4,7)(6,8)(9,13,14,11)(10,15,17,18)(12,22,21,19)(16,23,20,24)(25,29,31,36)(26,33,34,39)(27,28,37,40)(30,38,35,32);(1,3)(2,6)(4,8)(5,7)(9,10)(11,15)(12,20)(13,18)(14,17)(16,21)(19,24)(22,23)(25,30)(26,27)(28,39)(29,32)(31,35)(33,40)(34,37)(36,38)

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=h,d^2=gh,e^2=g,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, 8609887);