#511724

order	512
nilpotency class	4
isoclinism	isoclinic class of #511577
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

367 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 367 conjugacy classes of subgroups — too many to draw. Only its 128 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 418920);