#232110

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	C₂²×D₄	C₂²×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×C₄⋊C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

575 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=ef,d^2=e,f^2=h,g^2=i,b^a=be,c^a=cf,f^a=fh,d^b=dei,d^c=dhi

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, 139306);