#112869

order	512
nilpotency class	3
isoclinism	isoclinic class of #112861
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	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

897 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 20065);