#122097

order	512
nilpotency class	3
isoclinism	isoclinic class of #120186
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#391
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1446
prime radical	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

637 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=fh,e^2=i,g^2=h,b^a=bd,c^a=ce,d^a=dh,f^a=fi,c^b=cf,d^b=di,g^b=gh,d^c=di,g^c=gi

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, 29293);