#253901

order	512
nilpotency class	3
isoclinism	isoclinic class of #253891
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3205	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

794 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:

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

Polycyclic presentation:

2^9|b^2=eh,d^2=i,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=ghi,c^b=cg,f^b=fi,g^b=gi,e^c=eh,f^c=fi,g^c=ghi,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, 161097);