#203945

order	512
nilpotency class	3
isoclinism	isoclinic class of #19384
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#389
prime radical	C₂⁴	C₂²⋊Q₈
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

622 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)(3,7)(4,6)(5,9)(8,12)(10,14)(11,13)(15,16)(17,18,22,26,23,27,21,31)(19,24,28,32,29,25,20,30);(2,6)(4,12)(7,13)(8,9)(11,16)(14,15)(17,19,22,28,23,29,21,20)(18,24,26,32,27,25,31,30);(1,3)(2,7)(4,11)(5,10)(6,13)(8,15)(9,14)(12,16)(17,20,21,29,23,28,22,19)(18,25,26,30,27,24,31,32);(3,10)(7,14)(11,16)(13,15)

Polycyclic presentation:

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

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, 111141);