#7646562

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	#37845	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4423 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=gi,d^2=g,e^2=i,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, 7553758);