#232005

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

572 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)(11,18)(12,15)(13,17)(14,21)(16,20)(19,23)(22,24)(25,26,30,34,29,32,39,33)(27,35,28,37,36,31,38,40);(1,2)(3,7)(4,8)(5,6)(11,19)(14,22)(18,23)(21,24)(25,27,30,28,29,36,39,38)(26,31,34,40,32,35,33,37);(1,3,5,4)(2,7,6,8)(9,11,20,24,13,19,12,21)(10,14,15,23,17,22,16,18)(25,28,30,36,29,38,39,27)(26,31,33,37,32,35,34,40);(1,4,5,3)(2,7,6,8)(9,12,13,20)(10,15,17,16)(11,21,19,24)(14,23,22,18)(27,36)(28,38)(31,35)(37,40)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=efh,d^2=eh,f^2=h,g^2=i,b^a=be,c^a=cf,f^a=fh,c^b=ch,d^b=de,d^c=di

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, 139201);