#112193

order	512
nilpotency class	3
isoclinism	isoclinic class of #112191
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1438
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	#60799	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

898 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:

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

Polycyclic presentation:

2^9|a^2=g,d^2=i,e^2=h,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,f^a=fhi,c^b=cf,d^b=di,f^b=fh,g^b=ghi,d^c=dhi,e^c=eh,f^c=fh,g^c=gh

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 19389);