#7657236

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

4366 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)(10,16)(12,18)(17,19)(20,22)(25,26)(27,37)(28,33)(29,34)(30,35)(31,36)(32,38)(39,40);(1,2)(3,7)(4,8)(5,6)(9,10,15,16)(11,19,13,17)(12,14,18,24)(20,23,22,21)(25,27,31,38)(26,32,36,37)(28,29,35,39)(30,34,33,40);(9,11,15,13)(10,17,16,19)(12,20,18,22)(14,21,24,23)(25,27,31,38)(26,32,36,37)(28,29,35,39)(30,34,33,40);(1,3,5,4)(2,7,6,8)(9,12,15,18)(10,14,16,24)(11,20,13,22)(17,21,19,23)(25,28,31,35)(26,33,36,30)(27,29,38,39)(32,40,37,34);(1,4,5,3)(2,8,6,7)(9,13,15,11)(10,17,16,19)(12,22,18,20)(14,21,24,23)(25,29)(26,34)(27,35)(28,38)(30,37)(31,39)(32,33)(36,40);(9,14)(10,18)(11,21)(12,16)(13,23)(15,24)(17,22)(19,20)(25,30,31,33)(26,35,36,28)(27,34,38,40)(29,32,39,37)

Polycyclic presentation:

2^9|b^2=hi,c^2=hi,d^2=g,e^2=gh,f^2=h,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, 7564432);