#7642932

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

4316 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=h,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, 7550128);