#273529

order	512
nilpotency class	3
isoclinism	isoclinic class of #273435
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂³	D₄²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

783 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)(3,11)(4,13)(5,15)(6,8)(7,12)(9,14)(10,16)(18,24,28,27)(19,29)(20,34,35,36)(25,30,40,39)(31,45,43,46)(32,33)(38,47)(44,48);(1,3,6,4)(2,7,8,9)(5,14,16,12)(10,13,15,11)(17,18)(19,30)(20,26)(21,27)(22,28)(23,24)(25,33)(29,39)(31,44)(32,40)(34,37)(35,42)(36,41)(38,46)(43,48)(45,47);(2,8)(7,9)(10,15)(11,13)(17,19,22,29)(18,25,28,40)(20,31,35,43)(21,33,23,32)(24,30,27,39)(26,38,42,47)(34,45,36,46)(37,44,41,48);(1,4,6,3)(2,9,8,7)(5,12,16,14)(10,11,15,13)(17,20,22,35)(18,26,28,42)(19,31,29,43)(21,36,23,34)(24,37,27,41)(25,38,40,47)(30,44,39,48)(32,45,33,46)

Polycyclic presentation:

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

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, 180725);