#7655331

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	#39963	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4372 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=ghi,d^2=g,e^2=gi,f^2=gi,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, 7562527);