#116273

order	512
nilpotency class	3
isoclinism	isoclinic class of #10378
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#393
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1439
prime radical	C₂⁴	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

609 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)(4,14)(5,8)(6,9)(7,11)(12,15)(13,16)(17,18,21,25,22,24,20,30)(19,26,29,32,28,23,27,31);(1,2)(3,10)(4,8)(5,14)(6,9)(7,12)(11,15)(13,16)(19,27,28,29)(23,31,26,32);(1,3,6,13)(2,7,9,15)(4,12,5,11)(8,16,14,10)(17,19,22,28)(18,23,24,26)(20,29,21,27)(25,32,30,31)

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=i,e^2=hi,f^2=i,g^2=h,b^a=bd,c^a=ce,e^a=ei,f^a=fh,c^b=cf,f^b=fhi,d^c=dh,e^c=ehi,f^c=fi,g^c=gh

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, 23469);