#7712985

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	#46366	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4344 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:

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

Polycyclic presentation:

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

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, 7620181);