#7714945

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

4349 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=g,d^2=i,e^2=i,f^2=h,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^b=di,d^c=dh,e^c=ei

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, 7622141);