#348024

order	512
nilpotency class	3
isoclinism	isoclinic class of #348018
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2847
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3459	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

829 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:

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

Polycyclic presentation:

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

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, 255220);