#7712578

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	#37546	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4508 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)(12,19)(16,21)(17,23)(22,24)(25,26,31,32)(27,29,36,33)(28,34,39,40)(30,37,38,35);(1,3,5,8)(2,6,4,7)(9,10)(11,15)(12,16)(13,14)(17,22)(18,20)(19,21)(23,24)(26,32)(27,36)(34,40)(35,37);(1,4,5,2)(3,7,8,6)(10,14)(15,20)(16,21)(22,24)(25,27,31,36)(26,33,32,29)(28,35,39,37)(30,34,38,40);(9,11,13,18)(10,15,14,20)(12,17,19,23)(16,22,21,24)(25,28)(26,34)(27,37)(29,38)(30,33)(31,39)(32,40)(35,36);(9,10)(11,15)(12,16)(13,14)(17,22)(18,20)(19,21)(23,24)(25,29,31,33)(26,27,32,36)(28,38,39,30)(34,37,40,35);(9,12,13,19)(10,16,14,21)(11,17,18,23)(15,22,20,24)(25,30,31,38)(26,35,32,37)(27,34,36,40)(28,33,39,29)

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=g,d^2=hi,e^2=h,f^2=i,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^c=dh,e^c=ei

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, 7619774);