#116223

order	512
nilpotency class	3
isoclinism	isoclinic class of #10512
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#394
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1439
prime radical	C₂⁴	#123
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

592 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:

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

Polycyclic presentation:

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

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, 23419);