#8701385

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	#45106	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4405 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=g,d^2=ghi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^d=fgi,f^e=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, 8608581);