#7654165

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	C₂²×#41	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4373 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)(11,18)(12,17)(13,16)(14,19)(15,22)(20,23)(21,24)(25,26,31,35)(27,36,38,32)(28,33,39,29)(30,34,40,37);(1,2,4,5)(3,6,7,8)(9,11)(10,15)(12,20)(13,21)(14,22)(16,23)(17,24)(18,19)(25,27,31,38)(26,32,35,36)(28,37,39,34)(29,30,33,40);(9,12,14,13)(10,16,19,17)(11,20,22,21)(15,23,18,24)(25,28)(26,33)(27,34)(29,35)(30,32)(31,39)(36,40)(37,38);(1,3,4,7)(2,6,5,8)(9,13,14,12)(10,17,19,16)(11,21,22,20)(15,24,18,23)(25,29,31,33)(26,28,35,39)(27,30,38,40)(32,37,36,34);(9,11)(10,18)(12,20)(13,21)(14,22)(15,19)(16,24)(17,23)(25,28)(26,33)(27,37)(29,35)(30,36)(31,39)(32,40)(34,38);(9,10)(11,15)(12,16)(13,17)(14,19)(18,22)(20,23)(21,24)(25,30)(26,34)(27,33)(28,32)(29,38)(31,40)(35,37)(36,39)

Polycyclic presentation:

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

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, 7561361);