#6687775

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682440
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	#39382	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1740 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)(21,22,25,27)(23,31,32,47)(24,36,37,29)(26,40,33,30)(28,42,41,35)(34,49,38,46)(39,45,48,51)(43,50,44,52);(1,2)(3,4)(5,7)(6,11)(8,16)(9,18)(10,17)(12,19)(13,20)(14,15)(22,27)(23,32)(24,37)(30,40)(35,42)(39,48)(43,44)(46,49);(2,4)(5,8,10,9)(6,12,14,13)(7,16,17,18)(11,19,15,20)(21,23,33,42,25,32,26,35)(22,28,40,47,27,41,30,31)(24,38,48,50,37,34,39,52)(29,44,51,46,36,43,45,49);(5,8,10,9)(6,13,14,12)(7,16,17,18)(11,20,15,19)(21,24)(22,29)(23,34)(25,37)(26,39)(27,36)(28,43)(30,45)(31,46)(32,38)(33,48)(35,50)(40,51)(41,44)(42,52)(47,49);(2,4)(5,9,10,8)(6,13,14,12)(7,16,17,18)(11,19,15,20)(24,37)(29,36)(34,38)(39,48)(43,44)(45,51)(46,49)(50,52)

Polycyclic presentation:

2^9|a^2=f,c^2=fg,d^2=fi,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,d^c=di,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6594971);