#7654643

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	#2678	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4410 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:

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

Polycyclic presentation:

2^9|c^2=hi,d^2=g,e^2=gi,f^2=i,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, 7561839);