#118038

order	512
nilpotency class	3
isoclinism	isoclinic class of #118036
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1439
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#124
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

590 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:

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

Polycyclic presentation:

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

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, 25234);