#7654217

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

4481 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)(11,17)(12,22)(13,15)(14,18)(16,20)(19,21)(23,24)(25,26)(27,34)(28,32)(29,33)(30,35)(31,38)(36,40)(37,39);(1,2)(3,6)(4,7)(5,8)(9,11,16,23)(10,17,20,24)(12,21,14,13)(15,22,19,18)(25,27)(26,31)(28,39)(29,37)(30,38)(32,40)(33,36)(34,35);(1,3,5,4)(2,6,8,7)(9,12,16,14)(10,18,20,22)(11,21,23,13)(15,24,19,17)(25,28,30,33)(26,32,35,29)(27,36,38,39)(31,37,34,40);(1,4,5,3)(2,7,8,6)(9,13)(10,19)(11,12)(14,23)(15,20)(16,21)(17,18)(22,24)(25,29,30,32)(26,33,35,28)(27,37,38,40)(31,36,34,39);(9,14,16,12)(10,18,20,22)(11,13,23,21)(15,24,19,17)(25,26)(27,31)(28,32)(29,33)(30,35)(34,38)(36,37)(39,40);(9,15,16,19)(10,13,20,21)(11,22,23,18)(12,24,14,17)(25,27)(26,34)(28,36)(29,37)(30,38)(31,35)(32,40)(33,39)

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=gh,d^2=g,e^2=h,f^2=h,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, 7561413);