#273512

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	#532	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

786 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:

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

Polycyclic presentation:

2^9|a^2=fi,c^2=hi,d^2=h,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, 180708);