#236020

order	512
nilpotency class	3
isoclinism	isoclinic class of #234079
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#120
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1222 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 143216);