#7720210

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4395 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)(10,15)(13,17)(16,22)(20,23)(25,26)(27,32)(28,30)(29,36)(31,35)(33,37)(34,38)(39,40);(1,2,5,6)(3,7,4,8)(9,10,14,15)(11,16,21,22)(12,13,18,17)(19,20,24,23)(25,27)(26,31)(28,32)(29,34)(30,35)(33,38)(36,39)(37,40);(10,15)(12,18)(16,22)(19,24)(25,27,30,35)(26,31,28,32)(29,34,37,40)(33,38,36,39);(1,3,5,4)(2,7,6,8)(9,11,14,21)(10,16,15,22)(12,19,18,24)(13,20,17,23)(25,27)(26,32)(28,31)(29,34)(30,35)(33,39)(36,38)(37,40);(1,4,5,3)(2,8,6,7)(9,12,14,18)(10,17,15,13)(11,19,21,24)(16,23,22,20)(25,28,30,26)(27,31,35,32)(29,36,37,33)(34,38,40,39);(9,13,14,17)(10,18,15,12)(11,20,21,23)(16,24,22,19)(25,29)(26,33)(27,34)(28,36)(30,37)(31,38)(32,39)(35,40)

Polycyclic presentation:

2^9|b^2=gh,c^2=h,d^2=gh,e^2=g,f^2=hi,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, 7627406);