#189043

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	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

877 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=e,d^2=h,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, 96239);