#183000

order	512
nilpotency class	3
isoclinism	isoclinic class of #17603
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)(4,8)(5,9)(6,10)(7,13)(12,16)(14,15)(17,18,21,26,23,28,22,27)(19,29,30,32,31,20,25,24);(1,3,5,12,6,14,4,13)(2,7,8,15,10,16,9,11)(17,19,22,25,23,31,21,30)(18,24,27,20,28,32,26,29);(1,2)(3,7)(4,9)(5,8)(6,10)(11,13)(12,15)(14,16)(17,20,21,24,23,29,22,32)(18,25,26,19,28,30,27,31);(1,4,6,5)(2,8,10,9)(3,12,14,13)(7,11,16,15)(17,21,23,22)(18,26,28,27)(19,25,31,30)(20,32,29,24)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 90196);