#112200

order	512
nilpotency class	3
isoclinism	isoclinic class of #112199
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	#3444	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

865 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)(3,9,11,15)(4,7)(5,8)(6,10,14,12)(13,16)(18,23)(19,27)(20,21)(25,26)(28,31)(29,30);(1,3)(2,6)(4,12)(5,11)(7,15)(8,14)(9,16)(10,13)(17,18)(19,28)(20,25)(21,23)(22,26)(24,27)(29,32)(30,31);(4,13)(6,14)(7,16)(10,12)(17,19,22,30)(18,24,26,32)(20,27,21,29)(23,28,25,31)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 19396);