#7743457

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
prime radical	C₂⁴	C₂⁵
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3970 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=i,d^2=i,e^2=hi,f^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dhi,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, 7650653);