#7645730

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

4222 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)(9,10,15,17)(11,14,13,23)(12,19,22,16)(18,24,20,21)(25,26,30,31)(27,29,34,32)(28,36,38,33)(35,39,40,37);(1,2)(3,6)(4,5)(7,8)(9,11,15,13)(10,14,17,23)(12,18,22,20)(16,21,19,24)(26,31)(27,34)(33,36)(35,40);(9,12)(10,16)(11,18)(13,20)(14,21)(15,22)(17,19)(23,24)(25,27,30,34)(26,32,31,29)(28,35,38,40)(33,39,36,37);(9,13,15,11)(10,14,17,23)(12,20,22,18)(16,21,19,24)(25,28,30,38)(26,33,31,36)(27,35,34,40)(29,37,32,39);(1,3,4,7)(2,6,5,8)(9,14)(10,13)(11,17)(12,21)(15,23)(16,20)(18,19)(22,24)(25,29,30,32)(26,34,31,27)(28,37,38,39)(33,40,36,35);(9,12)(10,16)(11,18)(13,20)(14,21)(15,22)(17,19)(23,24)

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7552926);