#290040

order	512
nilpotency class	3
isoclinism	isoclinic class of #290011
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	C₂²×D₈	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

592 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 197236);