#7656922

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	#34589	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4269 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:

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

Polycyclic presentation:

2^9|b^2=h,c^2=ghi,d^2=g,e^2=i,f^2=i,b^a=bg,c^a=ch,d^a=di,c^b=ci,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, 7564118);