#8925581

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

4262 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)(11,17)(15,22)(18,20)(23,24)(25,26,30,27)(28,33,36,31)(29,32,37,34)(35,39,40,38);(1,2,6,8)(3,5,7,4)(9,10)(11,15)(12,19)(13,14)(16,21)(17,22)(18,24)(20,23)(25,27,30,26)(28,33,36,31)(29,32,37,34)(35,38,40,39);(1,3,6,7)(2,4,8,5)(10,14)(15,22)(16,19)(23,24)(25,28)(26,31)(27,33)(29,35)(30,36)(32,38)(34,39)(37,40);(1,4,6,5)(2,3,8,7)(9,11)(10,15)(12,20)(13,17)(14,22)(16,24)(18,21)(19,23);(1,2)(3,4)(5,7)(6,8)(9,12)(10,16)(11,18)(13,21)(14,19)(15,23)(17,20)(22,24)(25,29,30,37)(26,32,27,34)(28,35,36,40)(31,38,33,39);(1,5,6,4)(2,7,8,3)(9,10)(11,15)(12,16)(13,14)(17,22)(18,23)(19,21)(20,24)(25,28)(26,31)(27,33)(29,35)(30,36)(32,38)(34,39)(37,40)

Polycyclic presentation:

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

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, 8832777);