#251805

order	512
nilpotency class	3
isoclinism	isoclinic class of #251800
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	#2404
prime radical	#3455	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

773 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 159001);