#193854

order	512
nilpotency class	3
isoclinism	isoclinic class of #24464
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#386
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1277 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,d^2=h,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,f^a=fh,e^b=eh,f^b=fh,g^b=gh,d^c=di,e^c=eh,g^c=gh

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 101050);