#183148

order	512
nilpotency class	3
isoclinism	isoclinic class of #3264
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄²	C₂²×D₄
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×C₄×C₂²⋊C₄
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄² ◁ C₂×#431 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

991 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 90344);