#193851

order	512
nilpotency class	3
isoclinism	isoclinic class of #23876
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	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1140 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=h,d^2=h,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=dh,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

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 101047);