#7712452

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

Polycyclic presentation:

2^9|a^2=h,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, 7619648);