#268723

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	C₂×#41	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

764 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=hi,d^2=h,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, 175919);