#115593

order	512
nilpotency class	3
isoclinism	isoclinic class of #10192
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

847 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 22789);