#411224

order	512
nilpotency class	3
isoclinism	isoclinic class of #393881
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

976 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=eg,b^2=gh,c^2=gi,d^2=f,e^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,e^b=ei

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, 318420);