#6559040

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535815
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#127	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1943 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:

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

Polycyclic presentation:

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

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, 6466236);