#173664

order	512
nilpotency class	3
isoclinism	isoclinic class of #173481
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	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

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

normal subgroups only

This group has 691 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) = 512

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 80860);