#7766561

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	#36374	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4383 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:

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

Polycyclic presentation:

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

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, 7673757);