#189049

order	512
nilpotency class	3
isoclinism	isoclinic class of #188968
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#380
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3444	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

801 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=e,d^2=hi,e^2=h,f^2=h,g^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^b=di,f^c=fh,g^c=ghi

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, 96245);