#6436382

order	512
nilpotency class	3
isoclinism	isoclinic class of #6433042
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

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 ◁ C₂×#376 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1986 conjugacy classes of subgroups; only the 618 normal subgroups are shown.

normal subgroups only

This group has 1986 conjugacy classes of subgroups — too many to draw. Only its 618 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

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

Polycyclic presentation:

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

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, 6343578);