#7655102

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	#2431	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4422 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=i,d^2=h,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, 7562298);