#7726979

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

4339 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)(9,10,14,12)(11,17,19,15)(13,16,21,20)(18,23,24,22)(25,26,31,28)(27,34,30,32)(29,36,38,33)(35,39,37,40);(1,2)(3,5)(4,6)(7,8)(9,11)(10,15)(12,17)(13,18)(14,19)(16,22)(20,23)(21,24)(27,30)(32,34)(35,37)(39,40);(1,3,4,7)(2,5,6,8)(10,12)(15,17)(16,20)(22,23)(25,27,31,30)(26,32,28,34)(29,37,38,35)(33,40,36,39);(9,12,14,10)(11,17,19,15)(13,20,21,16)(18,23,24,22)(25,28,31,26)(27,32,30,34)(29,36,38,33)(35,39,37,40);(9,11)(10,15)(12,17)(13,18)(14,19)(16,22)(20,23)(21,24)(25,29,31,38)(26,33,28,36)(27,35,30,37)(32,39,34,40);(9,13,14,21)(10,16,12,20)(11,18,19,24)(15,22,17,23)(25,30,31,27)(26,32,28,34)(29,37,38,35)(33,39,36,40)

Polycyclic presentation:

2^9|a^2=gi,c^2=gh,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, 7634175);