#240971

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	C₂×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

893 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)(3,11)(4,6)(5,9)(7,10)(8,13)(12,15)(14,16)(19,27)(20,21)(23,31)(24,25)(28,32)(29,30);(1,3,7,15)(2,8,10,16)(4,11,9,12)(5,13,6,14)(17,18,22,26)(19,28,29,31)(20,25,21,24)(23,27,32,30);(1,4,6,2,7,9,5,10)(3,12,13,8,15,11,14,16)(17,19)(18,23)(20,30)(21,27)(22,29)(24,28)(25,31)(26,32);(1,5,7,6)(2,9,10,4)(3,13,15,14)(8,12,16,11)(19,29)(23,32)(27,30)(28,31)

Polycyclic presentation:

2^9|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, 148167);