#228328

order	512
nilpotency class	3
isoclinism	isoclinic class of #23860
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

591 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)(11,17)(12,14)(13,21)(15,19)(16,20)(18,24)(22,23)(25,26,30,33,31,28,29,38)(27,32,37,39,35,34,36,40);(1,2)(3,6)(4,5)(7,8)(10,14)(15,23)(16,21)(17,24)(25,27,30,37,31,35,29,36)(26,32,33,39,28,34,38,40);(1,3,4,7)(2,6,5,8)(9,11)(10,15)(12,18)(13,22)(14,23)(16,24)(17,21)(19,20)(25,28,31,26)(27,34,35,32)(29,33,30,38)(36,39,37,40);(2,5)(6,8)(11,18)(15,23)(17,24)(19,22)(27,35)(32,34)(36,37)(39,40)

Polycyclic presentation:

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

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, 135524);