#232860

order	512
nilpotency class	3
isoclinism	isoclinic class of #10212
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#392
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#392
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1292 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|d^2=i,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fi,g^a=gi,c^b=cg,e^b=eh,g^b=ghi,e^c=ei,f^c=fi,g^c=ghi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 140056);