#484971

order	512
nilpotency class	4
isoclinism	isoclinic class of #484212
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	C₂×#466	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

723 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 723 conjugacy classes of subgroups — too many to draw. Only its 138 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)(17,18,23,26)(19,30,33,45)(20,24,37,40)(21,29,22,28)(25,38,42,32)(27,36,41,34)(31,44,35,43)(39,47,46,48);(1,3)(2,6)(4,9)(5,10)(7,12)(8,13)(11,16)(14,15)(18,24)(20,34)(21,22)(25,39)(26,40)(27,29)(28,41)(30,44)(31,47)(32,38)(35,48)(36,37)(42,46)(43,45);(1,4,5,11)(2,7,8,14)(3,9,10,16)(6,12,13,15)(17,19)(18,25)(20,35)(21,38)(22,32)(23,33)(24,39)(26,42)(27,44)(28,45)(29,30)(31,37)(34,48)(36,47)(40,46)(41,43);(18,26)(24,40)(25,42)(27,41)(28,29)(30,45)(39,46)(43,44)

Polycyclic presentation:

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

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, 392167);