#8422188

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4238 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=gh,e^2=h,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, 8329384);