#193857

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

1231 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)(3,13,15,16)(4,9,8,14)(5,10)(6,11)(7,12)(17,18,23,25,22,19,21,26)(20,24,30,31,29,27,28,32);(1,3,5,14,7,15,6,9)(2,8,10,13,12,4,11,16)(17,19,22,18)(20,27,29,24)(21,26,23,25)(28,32,30,31);(1,4,7,8)(2,9,12,14)(3,11,15,10)(5,16,6,13)(17,20,22,29)(18,24,19,27)(21,30,23,28)(25,32,26,31);(1,5,7,6)(2,10,12,11)(3,14,15,9)(4,16,8,13)(17,21,22,23)(18,25,19,26)(20,28,29,30)(24,31,27,32)

Polycyclic presentation:

2^9|a^2=g,b^2=egh,c^2=h,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, 101053);