#273305

order	512
nilpotency class	3
isoclinism	isoclinic class of #273243
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	#121	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

760 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 180501);