#7726853

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	#45303	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4327 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=g,d^2=h,e^2=ghi,f^2=g,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, 7634049);