#8750140

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

4464 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=i,d^2=gh,e^2=h,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, 8657336);