#240856

order	512
nilpotency class	3
isoclinism	isoclinic class of #240845
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

907 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:

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

Polycyclic presentation:

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

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, 148052);