#7645980

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

4588 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)(13,21)(17,24)(19,23)(20,22)(25,26,30,33)(27,28,35,31)(29,37,38,32)(34,39,40,36);(1,3,5,8)(2,6,4,7)(9,10)(11,15)(12,16)(13,17)(14,18)(19,22)(20,23)(21,24)(25,27)(26,31)(28,33)(29,34)(30,35)(32,39)(36,37)(38,40);(9,11)(10,12)(13,19)(14,16)(15,18)(17,20)(21,23)(22,24)(25,27)(26,31)(28,33)(29,34)(30,35)(32,39)(36,37)(38,40);(1,2,5,4)(3,6,8,7)(9,11)(10,15)(12,18)(13,19)(14,16)(17,22)(20,24)(21,23)(25,28,30,31)(26,27,33,35)(29,36,38,39)(32,34,37,40);(9,12,14,15)(10,16,18,11)(13,20,21,22)(17,23,24,19)(25,27)(26,28)(29,34)(30,35)(31,33)(32,36)(37,39)(38,40);(1,4,5,2)(3,7,8,6)(9,13)(10,17)(11,19)(12,20)(14,21)(15,22)(16,23)(18,24)(25,29,30,38)(26,32,33,37)(27,34,35,40)(28,36,31,39)

Polycyclic presentation:

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

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, 7553176);