#8188191

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
prime radical	#2428	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4313 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,d^2=gi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8095387);