#7658186

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

4370 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=g,d^2=hi,e^2=gi,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, 7565382);