#288659

order	512
nilpotency class	3
isoclinism	isoclinic class of #288643
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2428
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#493	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

598 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:

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

Polycyclic presentation:

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

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, 195855);