#273258

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	#62170	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

795 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=hi,d^2=hi,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, 180454);