#7646984

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

4147 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)(11,12)(14,16)(19,20)(21,23)(25,26,29,32)(27,33,35,30)(28,31,36,38)(34,39,40,37);(1,2)(3,6)(4,7)(5,8)(9,10,15,18)(11,16,12,14)(13,17,22,24)(19,23,20,21)(25,27,29,35)(26,30,32,33)(28,34,36,40)(31,37,38,39);(1,3,5,4)(2,6,8,7)(9,11,15,12)(10,16,18,14)(13,19,22,20)(17,23,24,21)(25,27)(26,30)(28,34)(29,35)(31,37)(32,33)(36,40)(38,39);(9,12,15,11)(10,14,18,16)(13,19,22,20)(17,23,24,21)(25,26)(27,30)(28,31)(29,32)(33,35)(34,37)(36,38)(39,40);(9,13,15,22)(10,17,18,24)(11,19,12,20)(14,21,16,23);(1,4,5,3)(2,7,8,6)(9,14)(10,11)(12,18)(13,21)(15,16)(17,19)(20,24)(22,23)(25,28,29,36)(26,31,32,38)(27,34,35,40)(30,37,33,39)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=gh,d^2=hi,e^2=hi,f^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, 7554180);