#7658921

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

4650 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)(9,10,14,15)(11,18,20,16)(12,17,21,22)(13,19,23,24)(26,32)(28,38)(33,35)(36,39);(1,2)(3,7)(4,8)(5,6)(10,15)(12,21)(16,18)(19,24)(25,26)(27,35)(28,30)(29,33)(31,32)(34,38)(36,40)(37,39);(9,11)(10,16)(12,19)(13,17)(14,20)(15,18)(21,24)(22,23)(25,27,31,29)(26,33,32,35)(28,36,38,39)(30,37,34,40);(9,12,14,21)(10,17,15,22)(11,19,20,24)(13,18,23,16)(25,28)(26,34)(27,36)(29,39)(30,32)(31,38)(33,40)(35,37);(1,3,5,4)(2,7,6,8)(9,11)(10,18)(12,19)(13,22)(14,20)(15,16)(17,23)(21,24)(25,29,31,27)(26,33,32,35)(28,39,38,36)(30,37,34,40);(1,4,5,3)(2,8,6,7)(9,13,14,23)(10,19,15,24)(11,17,20,22)(12,18,21,16)(25,30,31,34)(26,28,32,38)(27,37,29,40)(33,36,35,39)

Polycyclic presentation:

2^9|a^2=h,c^2=i,d^2=h,e^2=gh,f^2=g,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, 7566117);