#8421685

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	C₂×#120	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4458 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=h,d^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,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, 8328881);