#220365

order	512
nilpotency class	3
isoclinism	isoclinic class of #220244
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#547	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

912 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 127561);