#508748

order	512
nilpotency class	4
isoclinism	isoclinic class of #508008
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

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

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

normal subgroups only

This group has 401 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) = 56
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 56 points:

(1,2)(3,6)(4,5)(7,8)(9,10,13,16)(11,17,19,14)(12,15,20,22)(18,23,24,21)(25,26,29,34)(27,36,39,32)(28,41,31,33)(30,35,43,51)(37,52,55,46)(38,47,45,49)(40,53,54,48)(42,56,44,50);(9,11,13,19)(10,14,16,17)(12,18,20,24)(15,21,22,23)(25,27,29,39)(26,32,34,36)(28,38,31,45)(30,37,43,55)(33,47,41,49)(35,46,51,52)(40,42,54,44)(48,56,53,50);(2,5)(6,8)(9,12,13,20)(10,15,16,22)(11,18,19,24)(14,21,17,23)(27,37)(28,31)(30,44)(32,46)(33,41)(35,50)(36,52)(38,40)(39,55)(42,43)(45,54)(47,53)(48,49)(51,56);(1,3,4,7)(2,6,5,8)(12,20)(15,22)(18,24)(21,23)(25,28,29,31)(26,33,34,41)(27,38,39,45)(30,42,43,44)(32,47,36,49)(35,50,51,56)(37,54,55,40)(46,48,52,53)

Polycyclic presentation:

2^9|a^2=e,b^2=e,c^2=ei,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, 415944);