#116246

order	512
nilpotency class	3
isoclinism	isoclinic class of #10382
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	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1439
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

587 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 23442);