#508754

order	512
nilpotency class	4
isoclinism	isoclinic class of #508669
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#3109
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂³	C₂×AGL(1,ℤ/8)
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 374 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) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 415950);