#7658307

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

4506 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7565503);