#233050

order	512
nilpotency class	3
isoclinism	isoclinic class of #233019
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1228 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 52
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 52 points:

(1,2)(3,4)(5,6)(7,14)(8,12)(9,17)(10,13)(11,15)(16,19)(18,20)(21,22,27,30)(23,33,37,44)(24,29,41,46)(25,32,26,31)(28,36,43,35)(34,45,50,52)(38,42,49,51)(39,48,40,47);(2,4)(21,23,27,37)(22,28,30,43)(24,38,41,49)(25,36,26,35)(29,45,46,52)(31,44,32,33)(34,39,50,40)(42,47,51,48);(5,7)(6,11)(8,15)(9,18)(10,16)(12,19)(13,20)(14,17)(21,24,27,41)(22,29,30,46)(23,34,37,50)(25,40,26,39)(28,42,43,51)(31,48,32,47)(33,45,44,52)(35,38,36,49);(5,8,10,9)(6,12,13,17)(7,15,16,18)(11,19,20,14)(22,30)(28,43)(29,46)(31,32)(33,44)(42,51)(45,52)(47,48)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=h,d^2=i,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=dh,e^a=eh,f^a=fi,g^a=gh,c^b=cg,e^b=eh,g^b=gh,e^c=eh,f^c=fi,g^c=gh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 140246);