#232054

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

587 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)(25,26,30,35,31,34,29,39)(27,32,36,40,37,33,28,38);(1,2)(3,7)(4,8)(5,6)(11,18)(14,23)(17,24)(19,22)(25,27,29,28,31,37,30,36)(26,32,39,38,34,33,35,40);(1,3,5,4)(2,7,6,8)(9,11,12,19,13,18,21,22)(10,14,15,17,16,23,20,24)(25,28,29,37,31,36,30,27)(26,33,35,38,34,32,39,40);(1,4,5,3)(2,7,6,8)(11,18)(14,23)(17,24)(19,22)(26,34)(32,33)(35,39)(38,40)

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,d^a=di,f^a=fh,c^b=ch,d^b=de,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, 139250);