#238329

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

743 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

Polycyclic presentation:

2^9|a^2=eh,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, 145525);