#7712344

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	#36492	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4400 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=h,d^2=h,e^2=g,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, 7619540);