#228709

order	512
nilpotency class	3
isoclinism	isoclinic class of #30396
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

574 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)(9,10,13,15)(11,18,20,24)(12,17,14,21)(16,22,23,19)(25,26,29,33,30,28,35,32)(27,34,36,39,37,31,38,40);(1,2)(3,6)(4,5)(7,8)(10,15)(16,23)(17,21)(18,24)(25,27,35,38,30,37,29,36)(26,31,32,39,28,34,33,40);(1,3,4,7)(2,6,5,8)(9,11)(10,16)(12,19)(13,20)(14,22)(15,23)(17,24)(18,21)(25,28)(26,30)(27,31)(29,32)(33,35)(34,37)(36,40)(38,39);(2,5)(6,8)(9,12,13,14)(10,17,15,21)(11,19,20,22)(16,24,23,18)(25,29,30,35)(26,32,28,33)(27,36,37,38)(31,39,34,40)

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,d^a=di,f^a=fh,c^b=ci,d^b=deh,d^c=di,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, 135905);