#236164

order	512
nilpotency class	3
isoclinism	isoclinic class of #234044
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	C₂×#121	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1251 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)(7,12,16,20)(8,13,9,11)(15,18,17,19)(21,22,27,32)(23,33,37,43)(24,38,39,46)(25,31,26,30)(28,36,44,35)(29,40,45,41)(34,48,49,42)(47,50,52,51);(1,2)(3,4)(6,11)(8,9)(12,18)(13,14)(15,17)(19,20)(21,23,27,37)(22,28,32,44)(24,34,39,49)(25,36,26,35)(29,42,45,48)(30,43,31,33)(38,52,46,47)(40,51,41,50);(2,4)(5,7)(6,12)(8,17)(9,15)(10,16)(11,18)(13,19)(14,20)(21,24)(22,29)(23,34)(25,41)(26,40)(27,39)(28,42)(30,38)(31,46)(32,45)(33,47)(35,51)(36,50)(37,49)(43,52)(44,48);(5,8,10,9)(6,13,14,11)(7,15,16,17)(12,18,20,19)(24,39)(29,45)(34,49)(38,46)(40,41)(42,48)(47,52)(50,51)

Polycyclic presentation:

2^9|a^2=h,b^2=i,d^2=hi,e^2=h,f^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,c^b=cg,d^b=dhi,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, 143360);