#7655551

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	#42074	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4485 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:

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

Polycyclic presentation:

2^9|a^2=gh,c^2=hi,d^2=i,e^2=gh,f^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, 7562747);