#8750410

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

4557 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=h,d^2=gi,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,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, 8657606);