#6569792

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×#3347	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2044 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)(23,32)(24,29)(25,30)(26,40)(27,31)(28,35)(33,47)(34,43)(36,42)(37,44)(38,45)(39,46)(41,48)(49,51)(50,52);(5,7)(6,10)(8,14)(9,13)(11,17)(12,16)(15,19)(18,20)(21,23,26,35,27,36,25,34)(22,28,30,32,31,43,40,42)(24,33,37,48,39,49,38,50)(29,41,45,47,46,52,44,51);(1,2,3,4)(7,13)(10,16)(14,19)(17,20)(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);(5,8)(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)(24,37,39,38)(28,32,43,42)(29,44,46,45)(33,48,49,50)(41,51,52,47);(8,15)(11,18)(14,19)(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=h,b^2=f,c^2=g,d^2=i,e^2=i,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, 6476988);