#7653980

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

4484 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)(11,17)(15,18)(19,23)(22,24)(25,26)(27,34)(28,32)(29,31)(30,33)(35,40)(36,38)(37,39);(1,3,5,8)(2,6,4,7)(9,10,14,12)(11,15,17,18)(13,16,21,20)(19,22,23,24)(25,27,30,29)(26,31,33,34)(28,35,37,36)(32,38,39,40);(9,11)(10,15)(12,18)(13,19)(14,17)(16,22)(20,24)(21,23)(25,26)(27,34)(28,32)(29,31)(30,33)(35,40)(36,38)(37,39);(1,2,5,4)(3,6,8,7)(9,11)(10,15)(12,18)(13,19)(14,17)(16,22)(20,24)(21,23)(25,28,30,37)(26,32,33,39)(27,35,29,36)(31,38,34,40);(9,12,14,10)(11,18,17,15)(13,16,21,20)(19,22,23,24)(25,26)(27,31)(28,32)(29,34)(30,33)(35,38)(36,40)(37,39);(1,4,5,2)(3,7,8,6)(9,13)(10,16)(11,19)(12,20)(14,21)(15,22)(17,23)(18,24)(25,29,30,27)(26,31,33,34)(28,36,37,35)(32,38,39,40)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7561176);