#7658999

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	#2619	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4256 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)(9,10,15,19)(11,20,14,16)(12,17,13,18)(21,24,22,23)(26,31)(27,35)(33,34)(36,37);(1,3,5,8)(2,6,4,7)(9,11,15,14)(10,16,19,20)(12,22,13,21)(17,24,18,23)(26,31)(32,38)(33,34)(39,40);(9,12,15,13)(10,17,19,18)(11,21,14,22)(16,23,20,24)(25,26)(27,32)(28,33)(29,34)(30,31)(35,38)(36,39)(37,40);(9,10)(11,20)(12,17)(13,18)(14,16)(15,19)(21,24)(22,23)(25,27,30,35)(26,32,31,38)(28,36,29,37)(33,39,34,40);(1,2,5,4)(3,6,8,7)(9,13,15,12)(10,18,19,17)(11,21,14,22)(16,23,20,24)(25,28,30,29)(26,33,31,34)(27,36,35,37)(32,39,38,40);(1,4,5,2)(3,7,8,6)(9,14,15,11)(10,16,19,20)(12,22,13,21)(17,23,18,24)(25,29,30,28)(26,34,31,33)(27,37,35,36)(32,40,38,39)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=hi,d^2=h,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=di,c^b=ci,d^b=dhi,f^e=fhi

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, 7566195);