#511577

order	512
nilpotency class	4
isoclinism	isoclinic class of #511577
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₄³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#41 ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 696 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) = 256

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

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

Polycyclic presentation:

2^9|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, 418773);