#484976

order	512
nilpotency class	4
isoclinism	isoclinic class of #484217
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#3050	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#424 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

708 conjugacy classes of subgroups; only the 140 normal subgroups are shown.

normal subgroups only

This group has 708 conjugacy classes of subgroups — too many to draw. Only its 140 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)(3,6,10,13)(4,7,11,14)(9,15,16,12)(18,24)(19,29)(20,32)(21,22)(25,41)(26,33)(27,39)(28,40)(30,37)(31,34)(35,36)(38,44)(42,47)(43,45);(1,3)(2,6)(4,9)(5,10)(7,12)(8,13)(11,16)(14,15)(17,18,23,28)(19,25,30,42)(20,33,36,39)(21,27,22,26)(24,35,40,32)(29,45,37,44)(31,47,34,41)(38,46,43,48);(1,4,5,11)(2,7,8,14)(3,9,10,16)(6,12,13,15)(17,19,23,30)(18,25,28,42)(20,34,36,31)(21,37,22,29)(24,38,40,43)(26,45,27,44)(32,48,35,46)(33,41,39,47);(19,30)(25,42)(29,37)(31,34)(38,43)(41,47)(44,45)(46,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=h,e^2=gi,f^2=i,g^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fi,g^a=gi,e^b=eg,f^b=fh,g^b=gi,d^c=di,e^c=eh,f^c=fi

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, 392172);