#273797

order	512
nilpotency class	3
isoclinism	isoclinic class of #273755
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	#62091	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

781 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)(3,10,12,16)(4,9)(5,7)(6,11,8,13)(14,15)(17,18,22,26,23,28,21,27)(19,29,31,24,33,44,32,40)(20,34,36,25,38,42,37,43)(30,39,47,45,46,48,35,41);(1,3)(2,6)(4,13)(5,12)(7,8)(9,16)(10,15)(11,14)(17,19,23,33)(18,24,28,40)(20,35,38,47)(21,32,22,31)(25,41,43,45)(26,44,27,29)(30,37,46,36)(34,48,42,39);(2,7)(4,14)(6,8)(11,13)(17,20,23,38)(18,25,28,43)(19,30,33,46)(21,36,22,37)(24,39,40,48)(26,34,27,42)(29,45,44,41)(31,35,32,47);(1,3)(2,8)(4,13)(5,12)(6,7)(9,10)(11,14)(15,16)(17,18)(19,24)(20,25)(21,27)(22,26)(23,28)(29,32)(30,39)(31,44)(33,40)(34,37)(35,41)(36,42)(38,43)(45,47)(46,48)

Polycyclic presentation:

2^9|a^2=fi,b^2=h,c^2=h,e^2=i,f^2=h,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,f^c=fh,g^c=gh,e^d=ei,f^d=fhi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 180993);