#7645741

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4223 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)(9,10,15,17)(11,14,13,23)(12,19,22,16)(18,24,20,21)(26,30)(27,33)(32,37)(34,39);(1,3,5,8)(2,6,4,7)(9,11,15,13)(10,14,17,23)(12,18,22,20)(16,21,19,24)(25,26,29,30)(27,31,33,36)(28,32,35,37)(34,38,39,40);(1,2,5,4)(3,6,8,7)(9,12)(10,16)(11,18)(13,20)(14,21)(15,22)(17,19)(23,24)(25,26)(27,31)(28,32)(29,30)(33,36)(34,38)(35,37)(39,40);(9,13,15,11)(10,14,17,23)(12,20,22,18)(16,21,19,24)(25,27,29,33)(26,31,30,36)(28,34,35,39)(32,38,37,40);(9,14)(10,13)(11,17)(12,21)(15,23)(16,20)(18,19)(22,24);(1,4,5,2)(3,7,8,6)(9,12)(10,16)(11,18)(13,20)(14,21)(15,22)(17,19)(23,24)(25,28,29,35)(26,32,30,37)(27,34,33,39)(31,38,36,40)

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=gi,d^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,e^d=ehi

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, 7552937);