#8678455

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

4646 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=gh,d^2=h,e^2=gh,f^2=gi,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, 8585651);