#238361

order	512
nilpotency class	3
isoclinism	isoclinic class of #238361
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#393
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

799 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 145557);