#347925

order	512
nilpotency class	3
isoclinism	isoclinic class of #347908
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2847
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3455	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

809 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)(3,10,12,15)(4,8,5,13)(7,14,16,11)(18,23)(19,28)(21,24)(26,37)(27,38)(29,44)(30,31)(32,42)(33,34)(36,40)(39,45)(43,46);(17,18,24,38,22,27,21,23)(19,25,28,40,31,41,30,36)(20,32,34,37,35,26,33,42)(29,45,43,47,46,39,44,48);(1,3)(2,7)(4,11)(5,14)(6,12)(8,15)(9,16)(10,13)(17,19)(18,25)(20,29)(21,30)(22,31)(23,36)(24,28)(26,39)(27,41)(32,45)(33,44)(34,43)(35,46)(37,47)(38,40)(42,48);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,16,10)(17,20,22,35)(18,26,27,32)(19,29,31,46)(21,33,24,34)(23,37,38,42)(25,39,41,45)(28,43,30,44)(36,47,40,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=eh,d^2=i,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fi,d^b=dh,f^c=fi,g^c=gh,f^d=fh

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 255121);