#236758

order	512
nilpotency class	3
isoclinism	isoclinic class of #236755
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
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1493 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 143954);