#273738

order	512
nilpotency class	3
isoclinism	isoclinic class of #273691
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	#3515	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

772 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=h,c^2=hi,d^2=h,e^2=i,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,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

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 180934);