#180728

order	512
nilpotency class	3
isoclinism	isoclinic class of #180581
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#2867	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

724 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 724 conjugacy classes of subgroups — too many to draw. Only its 126 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:

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

Polycyclic presentation:

2^9|b^2=g,c^2=hi,d^2=gh,e^2=g,f^2=i,b^a=be,c^a=cf,d^a=dgh,e^a=eg,f^a=fi,d^b=dhi,e^b=eg,f^b=fh,d^c=dg,e^c=eh,f^c=fi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 87924);