#7660910

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	#52280	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4295 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:

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

Polycyclic presentation:

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

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, 7568106);