#238029

order	512
nilpotency class	3
isoclinism	isoclinic class of #238028
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#124
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#393
prime radical	C₂⁴	#120
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

740 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=e,c^2=i,f^2=hi,g^2=h,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,d^b=dhi,f^b=fh,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, 145225);