#494079

order	512
nilpotency class	4
isoclinism	isoclinic class of #12019
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#106
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂²×#106
prime radical	C₂⁴	C₂×SD₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

664 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)(3,6)(4,5)(7,8)(9,10)(11,24,28,38)(12,30,29,40)(13,20)(14,34)(15,35)(16,36)(17,23)(18,26,39,27)(19,32,37,31)(21,25)(22,33);(2,5)(6,8)(9,11,15,27,16,29,33,31,17,28,14,26,13,12,25,32)(10,18,22,38,20,37,35,40,23,39,21,24,36,19,34,30);(1,3,4,7)(2,6,5,8)(9,12,14,31,16,11,25,26,17,29,15,32,13,28,33,27)(10,19,21,40,20,18,34,24,23,37,22,30,36,39,35,38);(9,13,17,16)(10,20,23,36)(11,12,28,29)(14,33,15,25)(18,37,39,19)(21,34,22,35)(24,30,38,40)(26,31,27,32)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 401275);