#511725

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

Polycyclic presentation:

2^9|a^2=efi,b^2=ei,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, 418921);