#273250

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	#62021	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

827 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)(3,11,12,8)(4,14,15,7)(5,9)(6,10)(13,16)(17,18)(19,24)(20,32)(21,23)(22,27)(25,34)(26,36)(28,40)(29,44)(30,37)(31,41)(33,42)(35,38)(39,46)(43,48)(45,47);(1,3)(2,7)(4,13)(5,15)(6,12)(8,9)(10,14)(11,16)(18,23)(19,28)(21,36)(25,38)(26,27)(29,43)(30,31)(32,42)(33,34)(37,40)(44,47)(45,46);(3,12)(5,13)(7,14)(9,16)(17,19,22,31)(18,24,27,41)(20,29,35,45)(21,30,36,28)(23,37,26,40)(25,39,42,48)(32,44,38,47)(33,43,34,46);(1,4,6,15)(2,8,10,11)(3,13,12,5)(7,9,14,16)(17,20)(18,25)(19,29)(21,34)(22,35)(23,38)(24,39)(26,32)(27,42)(28,43)(30,46)(31,45)(33,36)(37,47)(40,44)(41,48)

Polycyclic presentation:

2^9|a^2=f,c^2=h,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, 180446);