#274034

order	512
nilpotency class	3
isoclinism	isoclinic class of #273947
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	#3035	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

815 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 181230);