#237126

order	512
nilpotency class	3
isoclinism	isoclinic class of #237115
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#392
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#60940	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1608 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:

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

Polycyclic presentation:

2^9|a^2=hi,e^2=h,f^2=hi,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fhi,g^a=gh,c^b=cg,d^b=dhi,e^b=eh,f^b=fhi,g^b=gh,e^c=ei,f^c=fhi,g^c=gh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 144322);