#238322

order	512
nilpotency class	3
isoclinism	isoclinic class of #238317
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#393
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#120
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

802 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=e,c^2=i,d^2=i,f^2=hi,g^2=h,b^a=be,c^a=cf,f^a=fi,g^a=ghi,c^b=cg,f^b=fi,d^c=di,e^c=eh,f^c=fhi,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, 145518);