#6687727

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682532
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#2540	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1725 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=fg,d^2=f,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,e^c=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6594923);