#317909

order	512
nilpotency class	3
isoclinism	isoclinic class of #317904
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2537
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#532	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

542 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)(3,9)(4,14)(5,8)(6,11)(7,12)(10,16)(13,15)(17,18,21,25)(19,22,27,31)(20,23,28,24)(26,29,32,30);(2,8,12,11)(3,13,10,14)(4,15)(9,16)(17,19,21,27)(18,22,25,31)(20,26,28,32)(23,29,24,30);(1,3)(2,9)(4,11)(5,13)(6,14)(7,10)(8,15)(12,16)(18,23)(20,28)(22,29)(24,25)(26,32)(30,31);(1,4,6,9,7,15,5,16)(2,10,8,13,12,3,11,14)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 225105);