#278533

order	512
nilpotency class	3
isoclinism	isoclinic class of #278523
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2416
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3106	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

529 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 185729);