#8108270

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	#38120	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4604 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=gh,d^2=g,e^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8015466);