#6559048

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, prime radical	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1896 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 52
transitive degree	μ_t(G) = 512

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

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=g,d^2=hi,e^2=hi,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, 6466244);