#112143

order	512
nilpotency class	3
isoclinism	isoclinic class of #112133
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

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×#3217	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

1018 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)(17,18,21,25)(19,26)(20,22,30,24)(23,29)(27,31)(28,32);(2,7)(3,11)(4,5)(8,15)(10,16)(12,13)(18,22)(19,27)(20,30)(24,25)(28,29)(31,32);(1,3)(2,8)(4,13)(5,11)(6,12)(7,10)(9,15)(14,16)(17,19)(18,23)(20,28)(21,29)(22,31)(24,32)(25,26)(27,30)

Polycyclic presentation:

2^9|a^2=g,d^2=i,f^2=h,g^2=hi,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, 19339);