#511423

order	512
nilpotency class	4
isoclinism	isoclinic class of #509081
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 ◁ C₂×C₄×Q₈ ◁ C₂×C₈×Q₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 457 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) = 48
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=efi,b^2=i,c^2=i,d^2=ef,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=defi,c^b=cg,d^b=de,g^b=gh,h^b=hi,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, 418619);