#7656771

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	C₂⁵	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4284 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=hi,e^2=gh,f^2=h,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, 7563967);