#7654953

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	#35303	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4496 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)(9,10,15,17)(11,18,13,16)(12,21,23,14)(19,20,22,24)(25,26,30,28)(27,31,33,34)(29,32,37,36)(35,38,39,40);(1,3,5,8)(2,6,4,7)(9,11,15,13)(10,16,17,18)(12,19,23,22)(14,20,21,24)(27,33)(31,34)(35,39)(38,40);(9,11,15,13)(10,16,17,18)(12,19,23,22)(14,20,21,24)(25,27,30,33)(26,31,28,34)(29,35,37,39)(32,38,36,40);(9,12)(10,14)(11,19)(13,22)(15,23)(16,20)(17,21)(18,24);(1,4,5,2)(3,7,8,6)(9,13,15,11)(10,16,17,18)(12,22,23,19)(14,20,21,24)(25,28,30,26)(27,34,33,31)(29,32,37,36)(35,38,39,40);(9,14)(10,12)(11,20)(13,24)(15,21)(16,19)(17,23)(18,22)(25,29,30,37)(26,32,28,36)(27,35,33,39)(31,38,34,40)

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=hi,e^2=gi,f^2=i,b^a=bg,c^a=ch,d^a=dh,c^b=ci,f^e=fhi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7562149);