#289142

order	512
nilpotency class	3
isoclinism	isoclinic class of #289115
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)(3,8,14,16)(4,12,7,13)(5,10,15,9)(17,18,22,26,21,25,28,31)(19,27,29,24,20,23,30,32);(1,3,5,12,6,14,15,13)(2,7,9,8,11,4,10,16)(17,19,28,30,21,20,22,29)(18,23,31,24,25,27,26,32);(1,3)(2,8)(4,9)(5,12)(6,14)(7,10)(11,16)(13,15)(17,20,21,19)(18,24,25,32)(22,29,28,30)(23,26,27,31);(1,4,15,8,6,7,5,16)(2,3,10,13,11,14,9,12)

Polycyclic presentation:

2^9|a^2=fh,b^2=ef,c^2=i,d^2=ehi,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=dei,e^a=eh,c^b=cg,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, 196338);