#7714902

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

4321 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)(9,10)(11,15)(12,14)(13,20)(16,21)(17,18)(19,24)(22,23)(25,26,31,28)(27,34,29,32)(30,36,38,33)(35,39,37,40);(1,3,5,8)(2,6,4,7)(9,11,14,17)(10,15,12,18)(13,19,21,23)(16,22,20,24)(25,27,31,29)(26,32,28,34)(30,35,38,37)(33,39,36,40);(10,12)(11,17)(16,20)(19,23)(27,29)(32,34)(35,37)(39,40);(9,11)(10,15)(12,18)(13,19)(14,17)(16,22)(20,24)(21,23)(25,28,31,26)(27,32,29,34)(30,36,38,33)(35,39,37,40);(9,12,14,10)(11,18,17,15)(13,20,21,16)(19,24,23,22)(25,29,31,27)(26,32,28,34)(30,37,38,35)(33,39,36,40);(1,4,5,2)(3,7,8,6)(9,13)(10,16)(11,19)(12,20)(14,21)(15,22)(17,23)(18,24)(25,30,31,38)(26,33,28,36)(27,35,29,37)(32,39,34,40)

Polycyclic presentation:

2^9|a^2=g,b^2=gh,d^2=hi,e^2=i,f^2=g,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, 7622098);