#7712337

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	C₂⁴	C₂⁵
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4184 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=h,d^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, 7619533);