#508729

order	512
nilpotency class	4
isoclinism	isoclinic class of #508008
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

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

483 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 483 conjugacy classes of subgroups — too many to draw. Only its 132 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) = 48
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=i,c^2=e,d^2=fi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=di,c^b=cg,g^b=gh,h^b=hi,d^c=de,g^c=gh,h^c=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 415925);