#220454

order	512
nilpotency class	3
isoclinism	isoclinic class of D₈×#99
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	C₂²×D₄	C₂²×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

839 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 127650);