#7646906

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	#2428	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4287 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)(9,10,15,19)(11,16,21,24)(12,22,13,17)(14,23,20,18)(25,26,32,35)(27,36,30,29)(28,33,40,37)(31,38,39,34);(1,3,5,8)(2,6,4,7)(9,11,15,21)(10,16,19,24)(12,20,13,14)(17,23,22,18)(25,27,32,30)(26,29,35,36)(28,38,40,34)(31,37,39,33);(9,12,15,13)(10,17,19,22)(11,20,21,14)(16,23,24,18)(25,28)(26,33)(27,34)(29,39)(30,38)(31,36)(32,40)(35,37);(9,10)(11,16)(12,17)(13,22)(14,18)(15,19)(20,23)(21,24)(25,29,32,36)(26,27,35,30)(28,39,40,31)(33,34,37,38);(9,13,15,12)(10,17,19,22)(11,14,21,20)(16,23,24,18)(25,30,32,27)(26,29,35,36)(28,38,40,34)(31,33,39,37);(1,4,5,2)(3,7,8,6)(9,14)(10,18)(11,12)(13,21)(15,20)(16,17)(19,23)(22,24)(25,31,32,39)(26,34,35,38)(27,37,30,33)(28,36,40,29)

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=h,d^2=i,e^2=hi,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, 7554102);