#499651

order	512
nilpotency class	4
isoclinism	isoclinic class of #498791
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #376 ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

704 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 704 conjugacy classes of subgroups — too many to draw. Only its 170 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) = 76
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 76 points:

(5,6,8,9)(7,10,11,12)(13,14,17,22,36,48,60,53,19,24,38,50,18,23,37,49)(15,25,28,51,39,67,65,42,27,55,59,43,29,46,34,20)(16,30,31,52,40,68,66,45,33,63,64,44,32,47,35,21)(26,41,57,70,62,72,76,73,56,69,75,74,58,71,61,54);(1,2)(3,4)(6,9)(10,12)(13,15,19,27)(14,20,24,42)(16,26,33,56)(17,34,38,65)(18,39,36,29)(21,41,45,69)(22,46,50,67)(23,51,48,43)(25,53,55,49)(28,60,59,37)(30,54,63,73)(31,61,64,76)(32,62,40,58)(35,57,66,75)(44,72,52,71)(47,70,68,74);(2,4)(13,16,19,33)(14,21,24,45)(15,26,27,56)(17,35,38,66)(18,40,36,32)(20,41,42,69)(22,47,50,68)(23,52,48,44)(25,54,55,73)(28,61,59,76)(29,62,39,58)(30,53,63,49)(31,60,64,37)(34,57,65,75)(43,72,51,71)(46,70,67,74);(5,7,8,11)(6,10,9,12)(15,27)(20,42)(25,55)(26,56)(28,59)(29,39)(34,65)(41,69)(43,51)(46,67)(54,73)(57,75)(58,62)(61,76)(70,74)(71,72)

Polycyclic presentation:

2^9|a^2=e,b^2=i,c^2=i,d^2=efh,e^2=hi,f^2=hi,h^2=i,b^a=be,c^a=cf,c^b=cg,d^b=di,e^b=eh,f^b=fh,h^b=hi,e^c=eh,f^c=fh,h^c=hi

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, 406847);