#7654478

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4316 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)(9,10,15,14)(11,17,12,16)(13,18,21,22)(19,24,20,23)(25,26,30,27)(28,31,36,33)(29,32,38,34)(35,39,37,40);(1,2)(3,7)(4,8)(5,6)(9,11,15,12)(10,16,14,17)(13,19,21,20)(18,23,22,24)(26,27)(31,33)(32,34)(39,40);(1,3,5,4)(2,7,6,8)(9,11,15,12)(10,16,14,17)(13,19,21,20)(18,23,22,24)(25,27,30,26)(28,33,36,31)(29,34,38,32)(35,40,37,39);(9,12,15,11)(10,17,14,16)(13,20,21,19)(18,24,22,23);(1,3,5,4)(2,7,6,8)(9,13)(10,18)(11,19)(12,20)(14,22)(15,21)(16,23)(17,24)(25,28,30,36)(26,31,27,33)(29,37,38,35)(32,40,34,39);(1,4,5,3)(2,8,6,7)(9,14,15,10)(11,17,12,16)(13,18,21,22)(19,23,20,24)(25,29,30,38)(26,32,27,34)(28,35,36,37)(31,39,33,40)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=g,d^2=h,e^2=gh,f^2=g,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, 7561674);