#7657210

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	#42970	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4354 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)(9,10,16,15)(11,20,22,17)(12,23,14,18)(13,19,21,24)(28,36)(33,38)(35,37)(39,40);(1,2)(3,7)(4,8)(5,6)(9,11)(10,17)(12,13)(14,21)(15,20)(16,22)(18,19)(23,24)(25,26,31,30)(27,34,29,32)(28,33,36,38)(35,40,37,39);(9,12,16,14)(10,18,15,23)(11,13,22,21)(17,19,20,24)(25,27,31,29)(26,32,30,34)(28,35,36,37)(33,39,38,40);(1,3,5,4)(2,7,6,8)(9,13,16,21)(10,19,15,24)(11,12,22,14)(17,18,20,23)(25,28)(26,33)(27,35)(29,37)(30,38)(31,36)(32,39)(34,40);(1,3,5,4)(2,7,6,8)(9,14,16,12)(10,18,15,23)(11,21,22,13)(17,19,20,24)(25,29,31,27)(26,32,30,34)(28,37,36,35)(33,39,38,40);(1,4,5,3)(2,8,6,7)(9,15,16,10)(11,20,22,17)(12,23,14,18)(13,24,21,19)(25,30,31,26)(27,34,29,32)(28,38,36,33)(35,40,37,39)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=hi,d^2=g,e^2=gi,f^2=gi,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, 7564406);