#511734

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

407 conjugacy classes of subgroups; only the 124 normal subgroups are shown.

normal subgroups only

This group has 407 conjugacy classes of subgroups — too many to draw. Only its 124 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=i,c^2=i,d^2=f,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, 418930);