#8718183

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

4614 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gi,d^2=hi,e^2=g,f^2=h,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, 8625379);