#268715

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

787 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:

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

Polycyclic presentation:

2^9|a^2=fh,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, 175911);