#8456014

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	#40590	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4730 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=gi,d^2=h,e^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8363210);