#283623

order	512
nilpotency class	3
isoclinism	isoclinic class of #283579
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2424
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#41	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

632 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:

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

Polycyclic presentation:

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

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, 190819);