#228301

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	C₂×#121	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×C₂²⋊C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

634 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)(9,10,13,18)(11,19,21,15)(12,17,14,16)(20,24,22,23)(25,26,29,33,31,28,30,38)(27,32,35,39,37,34,36,40);(1,2)(3,6)(4,5)(7,8)(9,11)(10,15)(12,22)(13,21)(14,20)(16,23)(17,24)(18,19)(25,27,30,36,31,37,29,35)(26,32,38,40,28,34,33,39);(1,3,4,7)(2,6,5,8)(10,16)(12,14)(15,23)(17,18)(19,24)(20,22)(25,28,31,26)(27,34,37,32)(29,38,30,33)(35,40,36,39);(2,5)(6,8)(9,12,13,14)(10,17,18,16)(11,20,21,22)(15,23,19,24)(25,29,31,30)(26,33,28,38)(27,35,37,36)(32,39,34,40)

Polycyclic presentation:

2^9|a^2=g,b^2=gh,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, 135497);