#228297

order	512
nilpotency class	3
isoclinism	isoclinic class of #23848
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×C₂²⋊C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

594 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=eh,d^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,f^a=fh,d^b=de,d^c=dhi,f^c=fh

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 135493);