#195301

order	512
nilpotency class	3
isoclinism	isoclinic class of #18446
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1242 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egi,d^2=i,e^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=di,f^a=fhi,e^b=eh,f^b=fi,g^b=gh,d^c=dh,e^c=ei,f^c=fh,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 102497);