#115401

order	512
nilpotency class	3
isoclinism	isoclinic class of #10004
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#392
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1438
prime radical	#3444	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

906 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=h,e^2=i,f^2=h,b^a=bd,c^a=ce,d^a=dh,f^a=fi,c^b=cf,d^b=dh,f^b=fh,d^c=di,e^c=ei,f^c=fh,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 22597);