#7726972

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	#34876	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4331 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:

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

Polycyclic presentation:

2^9|a^2=gi,c^2=g,d^2=h,e^2=i,f^2=h,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7634168);