#240134

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×#121	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

814 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=e,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, 147330);