#240974

order	512
nilpotency class	3
isoclinism	isoclinic class of #240963
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#394
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#124
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

823 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:

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

Polycyclic presentation:

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

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, 148170);