#238317

order	512
nilpotency class	3
isoclinism	isoclinic class of #238317
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

901 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

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,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.

Relationships

maximal subgroup of	#72233844, #72233848, #72233849
maximal normal subgroup of	#72233844, #72233848, #72233849

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 145513);