#7714977

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

4301 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)(9,10,14,15)(11,16,18,13)(12,20,22,17)(19,21,24,23)(25,26)(27,33)(28,36)(29,30)(31,34)(32,37)(35,38)(39,40);(1,3,5,8)(2,6,4,7)(25,26,29,30)(27,31,34,33)(28,32,37,36)(35,38,40,39);(10,15)(11,18)(17,20)(19,24)(26,30)(27,34)(32,36)(35,40);(1,2,5,4)(3,6,8,7)(9,11)(10,16)(12,19)(13,15)(14,18)(17,23)(20,21)(22,24)(25,26)(27,31)(28,32)(29,30)(33,34)(35,38)(36,37)(39,40);(1,2,5,4)(3,6,8,7)(9,12,14,22)(10,17,15,20)(11,19,18,24)(13,21,16,23)(25,27,29,34)(26,31,30,33)(28,35,37,40)(32,38,36,39);(1,4,5,2)(3,7,8,6)(9,13,14,16)(10,11,15,18)(12,21,22,23)(17,19,20,24)(25,28)(26,32)(27,35)(29,37)(30,36)(31,38)(33,39)(34,40)

Polycyclic presentation:

2^9|a^2=gh,b^2=g,d^2=gi,e^2=ghi,f^2=gh,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, 7622173);