#268756

order	512
nilpotency class	3
isoclinism	isoclinic class of #268627
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3511	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

806 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:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=h,c^2=hi,d^2=i,e^2=i,f^2=hi,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,e^b=ei,f^b=fhi,g^b=gi,e^c=ehi,f^c=fhi,g^c=gi,e^d=ei,f^d=fhi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 175952);