#232913

order	512
nilpotency class	3
isoclinism	isoclinic class of #232905
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

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

1282 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|d^2=hi,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,d^b=dh,e^b=eh,g^b=ghi,d^c=dh,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, 140109);