#7712441

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	#37621	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4478 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)(11,14)(12,16)(13,15)(17,19)(18,22)(20,21)(23,24)(25,26)(27,28)(29,37)(30,31)(32,38)(33,36)(34,40)(35,39);(1,2,4,5)(3,6,7,8)(9,11)(10,14)(12,18)(13,19)(15,17)(16,22)(20,23)(21,24)(26,31)(27,33)(32,37)(34,39);(1,3,4,7)(2,6,5,8)(10,13)(11,17)(12,20)(22,24)(25,27,30,33)(26,28,31,36)(29,34,38,39)(32,35,37,40);(9,12,15,20)(10,16,13,21)(11,18,17,23)(14,22,19,24);(9,13,15,10)(11,19,17,14)(12,21,20,16)(18,24,23,22)(25,28)(26,27)(29,35)(30,36)(31,33)(32,34)(37,39)(38,40);(9,14,15,19)(10,17,13,11)(12,22,20,24)(16,23,21,18)(25,29,30,38)(26,32,31,37)(27,34,33,39)(28,35,36,40)

Polycyclic presentation:

2^9|b^2=ghi,c^2=g,d^2=h,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, 7619637);