#473447

order	512
nilpotency class	3
isoclinism	isoclinic class of #454375
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×C₄×C₄⋊C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1008 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=fgh,c^2=i,d^2=ef,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^c=de,g^c=gi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 380643);