#112137

order	512
nilpotency class	3
isoclinism	isoclinic class of #112133
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	C₂×#3213	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

930 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:

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

Polycyclic presentation:

2^9|a^2=g,d^2=i,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fhi,c^b=cf,d^b=di,f^b=fh,g^b=ghi,d^c=dhi,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, 19333);