#283679

order	512
nilpotency class	3
isoclinism	isoclinic class of #283643
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	#3511	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

645 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)(17,18,21,30,23,20,22,25)(19,26,31,27,29,32,28,24);(1,3,6,15,8,4,7,14)(2,9,12,13,5,10,11,16)(17,19,22,28,23,29,21,31)(18,24,25,32,20,27,30,26);(1,4,8,3)(2,10,5,9)(6,15,7,14)(11,16,12,13)(17,18,23,20)(19,27,29,24)(21,25,22,30)(26,28,32,31);(1,5)(2,8)(3,10)(4,9)(6,11)(7,12)(13,14)(15,16)(17,20)(18,23)(19,27)(21,25)(22,30)(24,29)(26,28)(31,32)

Polycyclic presentation:

2^9|a^2=fi,b^2=f,c^2=h,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,d^c=di,f^c=fh,g^c=ghi,e^d=eh

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, 190875);