#7643443

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

4305 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,d^2=gh,e^2=g,f^2=hi,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, 7550639);