#237958

order	512
nilpotency class	3
isoclinism	isoclinic class of #237956
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#393
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#120
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

910 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=e,c^2=i,f^2=hi,g^2=h,b^a=be,c^a=cf,f^a=fi,g^a=gh,c^b=cg,d^b=dh,e^c=eh,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, 145154);