#6496405

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470149
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	C₂×#449	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1761 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)(23,28)(24,35)(25,30)(26,39)(27,31)(29,36)(32,47)(33,42)(34,43)(37,45)(38,44)(40,46)(41,50)(48,51)(49,52);(1,2,3,4)(5,7)(6,10)(8,14)(9,13)(11,18)(12,17)(15,20)(16,19)(21,23)(22,28)(24,32)(25,38)(26,33)(27,34)(29,41)(30,44)(31,43)(35,47)(36,50)(37,49)(39,42)(40,48)(45,52)(46,51);(7,13)(10,17)(14,19)(18,20)(21,24)(22,29)(23,32)(25,36)(26,40)(27,37)(28,41)(30,45)(31,46)(33,48)(34,49)(35,39)(38,50)(42,47)(43,51)(44,52);(7,13)(10,17)(14,19)(18,20)(21,25,27,26)(22,30,31,39)(23,33,34,38)(24,36,37,40)(28,42,43,44)(29,45,46,35)(32,48,49,50)(41,47,51,52);(5,8,9,16)(6,11,12,15)(7,14,13,19)(10,18,17,20)(23,34)(28,43)(32,49)(33,38)(41,51)(42,44)(47,52)(48,50)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6403601);