#6482250

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470894
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#533 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1652 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 52 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=f,d^2=h,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=dh,e^c=ei,e^d=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6389446);