#7658606

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4083 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:

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

Polycyclic presentation:

2^9|a^2=gh,c^2=gh,d^2=g,e^2=gi,f^2=gi,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, 7565802);