#7719100

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4170 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=gi,d^2=g,e^2=i,f^2=h,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, 7626296);