#220459

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	C₂×#555	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

861 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)(4,5)(8,14)(9,13)(10,11)(15,16)(17,18,21,26,22,25,19,23)(20,28,27,32,29,24,30,31);(1,2,6,9)(3,8,12,15)(4,7,5,13)(10,14,11,16)(17,19,22,21)(18,23,25,26)(20,27,29,30)(24,31,28,32);(1,3,6,12)(2,8,9,15)(4,10,5,11)(7,14,13,16)(17,20)(18,24)(19,27)(21,30)(22,29)(23,31)(25,28)(26,32);(1,4)(3,10)(5,6)(7,13)(11,12)(14,16)(17,19,22,21)(18,23,25,26)(20,27,29,30)(24,31,28,32)

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=h,d^2=i,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, 127655);