#111168

order	512
nilpotency class	3
isoclinism	isoclinic class of #111151
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1437
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂×#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

651 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=efh,d^2=hi,e^2=h,f^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=gi,d^b=dh,e^b=ei,f^b=fi,d^c=di

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 18364);