#183036

order	512
nilpotency class	3
isoclinism	isoclinic class of #17597
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×C₄×C₂²⋊C₄
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#431 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1012 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)(3,11)(4,7)(5,10)(6,9)(8,14)(12,15)(13,16)(17,18,22,27,23,26,21,32)(19,20,29,24,28,31,25,30);(1,3,6,15,7,13,5,14)(2,8,10,16,4,12,9,11)(17,19,22,29,23,28,21,25)(18,24,27,31,26,30,32,20);(1,4,7,2)(3,12,13,8)(5,10,6,9)(11,14,16,15)(17,20,21,30,23,31,22,24)(18,25,32,28,26,29,27,19);(2,4)(3,13)(9,10)(14,15)(18,26)(19,28)(25,29)(27,32)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=fh,e^2=h,f^2=i,g^2=h,b^a=be,c^a=ch,d^a=dh,e^a=eh,g^a=ghi,d^b=dh,e^b=ei,d^c=dhi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 90232);