#7650952

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

4405 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7558148);