#7657891

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

4421 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=g,d^2=hi,e^2=i,f^2=i,b^a=bg,c^a=ch,d^a=di,c^b=ci,f^e=fhi

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, 7565087);