#6481584

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470003
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#537	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2879 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)(21,22,26,29)(23,31,34,46)(24,28,36,45)(25,38,40,30)(27,39,42,33)(32,47,35,49)(37,52,51,44)(41,50,43,48);(5,7)(6,10)(8,15)(9,13)(11,19)(12,17)(14,16)(18,20)(21,23,26,34)(22,27,29,42)(24,35,36,32)(25,39,40,33)(28,43,45,41)(30,46,38,31)(37,48,51,50)(44,47,52,49);(2,4)(7,13)(10,17)(14,15)(18,19)(21,24)(22,28)(23,32)(25,37)(26,36)(27,41)(29,45)(30,44)(31,47)(33,48)(34,35)(38,52)(39,50)(40,51)(42,43)(46,49);(5,8,9,16)(6,11,12,20)(7,14,13,15)(10,18,17,19)(21,24)(22,28)(23,32)(25,37)(26,36)(27,41)(29,45)(30,44)(31,47)(33,48)(34,35)(38,52)(39,50)(40,51)(42,43)(46,49);(6,12)(10,17)(11,20)(18,19)(22,29)(24,36)(27,42)(30,38)(31,46)(32,35)(37,51)(48,50)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6388780);