#342564

order	512
nilpotency class	3
isoclinism	isoclinic class of #334823
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

655 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=fh,c^2=i,d^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=defi,c^b=cg,d^b=de,g^b=gi,f^c=fh,g^c=gi,f^d=fh

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 249760);