#8696226

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	#2555	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4263 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)(3,6)(4,7)(5,8)(9,10,15,19)(11,16,20,22)(12,17,21,24)(13,14,18,23)(26,29)(27,35)(28,38)(34,40);(1,3)(2,4)(5,7)(6,8)(11,20)(12,21)(16,22)(17,24)(25,26,31,29)(27,36,35,32)(28,39,38,33)(30,34,37,40);(9,11,15,20)(10,16,19,22)(12,14,21,23)(13,17,18,24)(25,27)(26,32)(28,37)(29,36)(30,38)(31,35)(33,40)(34,39);(1,2,5,8)(3,6,7,4)(9,12)(10,17)(11,14)(13,22)(15,21)(16,18)(19,24)(20,23)(25,28,31,38)(26,33,29,39)(27,37,35,30)(32,40,36,34);(1,3)(2,6)(4,8)(5,7)(9,13)(10,14)(11,17)(12,22)(15,18)(16,21)(19,23)(20,24)(25,29,31,26)(27,36,35,32)(28,39,38,33)(30,40,37,34);(1,4,5,6)(2,7,8,3)(9,14)(10,18)(11,21)(12,20)(13,19)(15,23)(16,24)(17,22)(25,30,31,37)(26,34,29,40)(27,38,35,28)(32,39,36,33)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=hi,d^2=g,e^2=h,f^2=g,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, 8603422);