#7655777

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	#40661	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4468 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=g,e^2=h,b^a=bg,c^a=ch,d^a=di,c^b=ci,f^e=fhi

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, 7562973);