#7642935

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

4324 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)(9,10,15,12)(11,19,20,16)(13,17,22,23)(14,24,21,18)(25,26)(27,32)(28,30)(29,37)(31,34)(33,39)(35,38)(36,40);(1,2)(3,6)(4,7)(5,8)(9,11)(10,16)(12,19)(13,21)(14,22)(15,20)(17,24)(18,23)(25,27,31,28)(26,32,34,30)(29,35,39,36)(33,40,37,38);(11,20)(13,22)(16,19)(17,23)(25,28,31,27)(26,32,34,30)(29,36,39,35)(33,40,37,38);(1,3,5,4)(2,6,8,7)(9,12,15,10)(11,19,20,16)(13,23,22,17)(14,24,21,18)(25,29)(26,33)(27,35)(28,36)(30,38)(31,39)(32,40)(34,37);(9,13,15,22)(10,17,12,23)(11,21,20,14)(16,24,19,18)(25,30,31,32)(26,28,34,27)(29,38,39,40)(33,36,37,35);(1,4,5,3)(2,7,8,6)(9,14,15,21)(10,18,12,24)(11,22,20,13)(16,23,19,17)(25,29)(26,33)(27,35)(28,36)(30,38)(31,39)(32,40)(34,37)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=h,d^2=g,e^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,e^c=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, 7550131);