#7658215

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

4390 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)(11,16)(12,22)(13,18)(14,17)(15,19)(20,24)(21,23)(25,26,32,31)(27,36,34,29)(28,33,40,39)(30,35,38,37);(1,2)(3,7)(4,8)(5,6)(9,11,15,13)(10,16,19,18)(12,20,14,21)(17,23,22,24)(25,27)(26,29)(28,35)(30,33)(31,36)(32,34)(37,40)(38,39);(9,12,15,14)(10,17,19,22)(11,20,13,21)(16,23,18,24)(25,28)(26,33)(27,37)(29,38)(30,36)(31,39)(32,40)(34,35);(9,13,15,11)(10,18,19,16)(12,21,14,20)(17,24,22,23)(25,29,32,36)(26,34,31,27)(28,38,40,30)(33,35,39,37);(1,3,5,4)(2,7,6,8)(9,14,15,12)(10,17,19,22)(11,21,13,20)(16,23,18,24)(25,30,32,38)(26,35,31,37)(27,33,34,39)(28,36,40,29);(1,4,5,3)(2,8,6,7)(9,10)(11,16)(12,17)(13,18)(14,22)(15,19)(20,23)(21,24)(25,31,32,26)(27,36,34,29)(28,39,40,33)(30,35,38,37)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=h,d^2=hi,e^2=gh,f^2=g,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, 7565411);