#7726949

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	#40426	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4229 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=g,d^2=g,e^2=i,f^2=h,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=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, 7634145);