#7716110

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	#45404	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4400 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|b^2=h,c^2=h,d^2=h,e^2=gh,f^2=g,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,e^b=ei,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, 7623306);