#180613

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
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

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

normal subgroups only

This group has 851 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) = 128

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

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

Polycyclic presentation:

2^9|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, 87809);