#8757059

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

4279 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=i,d^2=g,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8664255);