#7641998

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	C₂⁴	C₂⁵
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4248 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)(9,10,13,16)(11,17,19,14)(12,15,20,22)(18,23,24,21)(25,26,30,28)(27,32,35,31)(29,36,37,33)(34,38,40,39);(1,2)(3,6)(4,7)(5,8)(12,20)(15,22)(18,24)(21,23)(25,27,30,35)(26,31,28,32)(29,34,37,40)(33,38,36,39);(1,3,5,4)(2,6,8,7)(9,10)(11,14)(12,15)(13,16)(17,19)(18,21)(20,22)(23,24)(25,27)(26,32)(28,31)(29,34)(30,35)(33,39)(36,38)(37,40);(9,11,13,19)(10,14,16,17)(12,18,20,24)(15,21,22,23);(9,12,13,20)(10,15,16,22)(11,18,19,24)(14,21,17,23)(25,28,30,26)(27,31,35,32)(29,36,37,33)(34,38,40,39);(1,4,5,3)(2,7,8,6)(9,10)(11,14)(12,15)(13,16)(17,19)(18,21)(20,22)(23,24)(25,29,30,37)(26,33,28,36)(27,34,35,40)(31,38,32,39)

Polycyclic presentation:

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

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, 7549194);