#7654531

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	#2856	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4357 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)(11,18)(12,17)(13,16)(14,19)(15,20)(21,23)(22,24)(25,26,30,28)(27,31,33,34)(29,32,37,36)(35,38,39,40);(1,3,5,8)(2,6,4,7)(9,11)(10,14)(12,21)(13,22)(15,19)(16,23)(17,24)(18,20)(27,33)(31,34)(35,39)(38,40);(1,2,5,4)(3,6,8,7)(9,12,15,13)(10,16,20,17)(11,21,19,22)(14,23,18,24)(25,27)(26,31)(28,34)(29,35)(30,33)(32,38)(36,40)(37,39);(9,13,15,12)(10,17,20,16)(11,22,19,21)(14,24,18,23);(1,4,5,2)(3,7,8,6)(9,11)(10,18)(12,21)(13,22)(14,20)(15,19)(16,24)(17,23)(25,28,30,26)(27,34,33,31)(29,32,37,36)(35,38,39,40);(9,14,15,18)(10,19,20,11)(12,23,13,24)(16,22,17,21)(25,29,30,37)(26,32,28,36)(27,35,33,39)(31,38,34,40)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=g,d^2=h,e^2=ghi,f^2=hi,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, 7561727);