#6686211

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667580
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	#392	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1721 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)(7,13)(8,12)(9,11)(10,15)(14,20)(16,18)(17,19)(21,22)(23,31)(24,29)(25,28)(26,30)(27,35)(32,39)(33,45)(34,44)(36,41)(37,43)(38,42)(40,48)(46,50)(47,51)(49,52);(5,7,9,15)(6,10,11,13)(8,16,17,14)(12,20,19,18)(21,23)(22,27)(24,33)(25,35)(26,32)(28,31)(29,40)(30,44)(34,38)(36,48)(37,46)(39,42)(41,45)(43,51)(47,49)(50,52);(1,2,3,4)(22,28)(23,32)(24,36)(26,38)(27,39)(31,44)(33,47)(34,35)(40,51)(43,52)(45,50)(46,48);(6,11)(10,13)(12,19)(18,20)(21,24,25,36)(22,29,28,41)(23,33,35,48)(26,37,38,49)(27,40,31,45)(30,43,42,52)(32,46,34,47)(39,50,44,51);(5,8,9,17)(6,12,11,19)(7,14,15,16)(10,18,13,20)(23,34)(26,38)(27,39)(30,42)(31,44)(32,35)(33,47)(37,49)(40,50)(43,52)(45,51)(46,48)

Polycyclic presentation:

2^9|b^2=fi,c^2=gi,d^2=i,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dfi,c^b=ch,e^b=efh,h^b=hi,d^c=di,h^c=hi,h^e=hi

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, 6593407);