#6569697

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535655
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2006 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)(5,6,9,12)(7,10,13,16)(8,11,15,18)(14,17,19,20)(21,22,27,31)(23,32,36,47)(24,29,38,45)(25,30,26,40)(28,34,42,35)(33,46,49,51)(37,43,39,44)(41,50,52,48);(5,7)(6,10)(8,14)(9,13)(11,17)(12,16)(15,19)(18,20)(21,23,25,34,27,36,26,35)(22,28,40,47,31,42,30,32)(24,33,39,50,38,49,37,48)(29,41,43,51,45,52,44,46);(1,2,3,4)(7,13)(10,16)(14,19)(17,20)(21,24)(22,29)(23,33)(25,39)(26,37)(27,38)(28,41)(30,44)(31,45)(32,46)(34,50)(35,48)(36,49)(40,43)(42,52)(47,51);(5,8)(6,11)(7,14)(9,15)(10,17)(12,18)(13,19)(16,20)(21,25,27,26)(22,30,31,40)(23,34,36,35)(24,37,38,39)(28,32,42,47)(29,43,45,44)(33,48,49,50)(41,51,52,46);(8,15)(11,18)(14,19)(17,20)(21,25,27,26)(22,30,31,40)(23,34,36,35)(24,37,38,39)(28,32,42,47)(29,43,45,44)(33,48,49,50)(41,51,52,46)

Polycyclic presentation:

2^9|a^2=hi,b^2=fi,c^2=g,d^2=i,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^c=di,e^c=ei,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, 6476893);