#7644062

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	#42260	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4165 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)(9,10,15,17)(11,18,13,16)(12,14,22,24)(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)(9,11,15,13)(10,16,17,18)(12,21,22,19)(14,20,24,23)(25,26,29,32)(27,30,35,33)(28,31,36,38)(34,37,40,39);(11,13)(16,18)(19,21)(20,23)(25,26)(27,30)(28,31)(29,32)(33,35)(34,37)(36,38)(39,40);(1,3,5,4)(2,6,8,7)(9,12)(10,14)(11,19)(13,21)(15,22)(16,23)(17,24)(18,20)(25,27,29,35)(26,30,32,33)(28,34,36,40)(31,37,38,39);(1,4,5,3)(2,7,8,6)(9,13,15,11)(10,16,17,18)(12,21,22,19)(14,23,24,20);(9,14)(10,12)(11,20)(13,23)(15,24)(16,21)(17,22)(18,19)(25,28,29,36)(26,31,32,38)(27,34,35,40)(30,37,33,39)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7551258);