#189264

order	512
nilpotency class	3
isoclinism	isoclinic class of #19279
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

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×#380
prime radical	C₂²×#41	Q₈
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

931 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 96460);