#6815290

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682380
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1652 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:

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

Polycyclic presentation:

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

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, 6722486);