#241008

order	512
nilpotency class	3
isoclinism	isoclinic class of #240995
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	#3444	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

810 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)(4,5)(6,9)(7,10)(8,14)(12,13)(15,16)(17,18)(19,28)(20,29)(21,25)(22,26)(23,27)(24,32)(30,31);(1,3,7,13)(2,8,10,16)(4,11,9,12)(5,15,6,14)(17,19,22,29)(18,20,26,28)(21,30,23,24)(25,32,27,31);(1,4,5,2,7,9,6,10)(3,12,14,8,13,11,15,16)(17,20,22,28)(18,24,26,30)(19,25,29,27)(21,31,23,32);(3,13)(8,16)(11,12)(14,15)(17,21,22,23)(18,25,26,27)(19,24,29,30)(20,31,28,32)

Polycyclic presentation:

2^9|b^2=i,c^2=e,d^2=hi,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,d^b=di,e^b=eh,f^b=fi,g^b=ghi,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, 148204);