#233749

order	512
nilpotency class	3
isoclinism	isoclinic class of #233734
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	C₂³×D₄	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1272 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 52
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,d^2=i,e^2=h,f^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fhi,c^b=cg,d^b=di,e^b=eh,f^b=fi,d^c=dh,e^c=ei,f^c=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, 140945);