#7645891

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4143 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=gi,d^2=g,e^2=g,f^2=h,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, 7553087);