#240135

order	512
nilpotency class	3
isoclinism	isoclinic class of #240131
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

865 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)(3,11,12,15)(4,7,5,9)(8,13,16,14)(17,18,21,25)(19,22,26,31)(20,28,30,24)(23,29,32,27);(2,7)(3,12)(4,5)(8,11)(9,10)(15,16)(17,19)(18,22)(20,27)(21,26)(23,28)(24,32)(25,31)(29,30);(1,3,4,13,6,12,5,14)(2,8,9,15,10,16,7,11)(17,19)(18,23)(20,29)(21,26)(22,28)(24,31)(25,32)(27,30);(1,4,6,5)(2,7,10,9)(3,13,12,14)(8,11,16,15)(19,26)(22,31)(23,32)(27,29)

Polycyclic presentation:

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

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, 147331);