#6497568

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470003
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1636 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)(7,10,13,17)(8,15,16,11)(14,20,19,18)(21,22)(23,32)(24,37)(25,29)(26,40)(27,31)(28,35)(30,42)(33,48)(34,47)(36,46)(38,45)(39,52)(41,44)(43,50)(49,51);(5,7)(6,10)(8,14)(9,13)(11,18)(12,17)(15,20)(16,19)(21,23,27,36)(22,28,31,44)(24,33,25,34)(26,41,42,35)(29,43,37,51)(30,46,40,32)(38,50,39,49)(45,48,52,47);(1,2,3,4)(7,13)(10,17)(14,19)(18,20)(21,24,27,25)(22,29,31,37)(23,33,36,34)(26,38,42,39)(28,43,44,51)(30,45,40,52)(32,47,46,48)(35,49,41,50);(7,13)(10,17)(14,19)(18,20)(21,25,27,24)(22,29,31,37)(23,34,36,33)(26,39,42,38)(28,43,44,51)(30,45,40,52)(32,47,46,48)(35,50,41,49);(5,8,9,16)(6,11,12,15)(7,14,13,19)(10,18,17,20)(24,25)(29,37)(33,34)(38,39)(43,51)(45,52)(47,48)(49,50)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=g,d^2=i,e^2=h,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,f^b=fi,e^c=ei,e^d=ei

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, 6404764);