#116890

order	512
nilpotency class	3
isoclinism	isoclinic class of #9930
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#392
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1439
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

652 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:

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

Polycyclic presentation:

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

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, 24086);