#183167

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

1036 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:

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

Polycyclic presentation:

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

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, 90363);