#263330

order	512
nilpotency class	3
isoclinism	isoclinic class of #263311
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	#3169	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

785 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)(3,7)(4,12)(5,14)(6,10)(8,13)(9,15)(11,16)(17,18)(19,27)(20,32)(21,35)(22,26)(23,30)(24,34)(25,28)(29,43)(31,39)(33,46)(36,41)(37,44)(38,42)(40,45)(47,48);(1,3,6,4)(2,7,10,12)(5,13,15,11)(8,9,16,14)(17,19,28,41,22,31,21,30)(18,23,35,39,26,36,25,27)(20,29,42,48,33,44,34,45)(24,37,46,47,38,43,32,40);(1,4,6,3)(2,8,10,16)(5,11,15,13)(7,14,12,9)(20,33)(23,36)(24,38)(27,39)(29,44)(32,46)(34,42)(45,48);(17,20,28,42,22,33,21,34)(18,24,35,46,26,38,25,32)(19,29,41,48,31,44,30,45)(23,37,39,47,36,43,27,40)

Polycyclic presentation:

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

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, 170526);