#238328

order	512
nilpotency class	3
isoclinism	isoclinic class of #238324
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

751 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2)(3,7)(4,9)(5,13)(6,10)(8,11)(12,16)(14,15)(17,18)(19,24)(20,32,34,43)(21,26)(22,37)(23,27)(25,35,42,36)(28,41,44,48)(29,39)(30,40)(31,46)(33,38,45,47);(1,3)(2,7)(4,11)(5,14)(6,12)(8,9)(10,16)(13,15)(17,19)(18,24)(20,33,34,45)(21,30)(22,31)(23,29)(25,41,42,48)(26,40)(27,39)(28,36,44,35)(32,47,43,38)(37,46);(1,3)(2,8)(4,11)(5,14)(6,12)(7,13)(9,16)(10,15)(17,20)(18,25)(19,28)(21,36)(22,35)(23,34)(24,38)(26,43)(27,42)(29,44)(30,33)(31,45)(32,37)(39,47)(40,48)(41,46);(1,4,6,5)(2,9,10,13)(3,11,12,14)(7,8,16,15)(19,29)(24,39)(28,44)(30,31)(33,45)(38,47)(40,46)(41,48)

Polycyclic presentation:

2^9|a^2=e,b^2=e,d^2=i,f^2=hi,g^2=h,b^a=be,c^a=cf,f^a=fi,g^a=ghi,c^b=cg,d^b=dh,f^b=fi,e^c=eh,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, 145524);