#263907

order	512
nilpotency class	3
isoclinism	isoclinic class of #263899
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2408
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

813 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|b^2=e,c^2=i,d^2=e,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,c^b=cg,d^b=dh,g^b=ghi,f^c=fi,g^c=ghi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 171103);