#176499

order	512
nilpotency class	3
isoclinism	isoclinic class of #176333
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

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

602 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 602 conjugacy classes of subgroups — too many to draw. Only its 128 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)(7,11)(12,16)(13,15)(17,18)(19,22)(20,23)(21,24)(25,28)(26,29)(27,30)(31,32)(33,34,39,35)(36,44,46,48)(37,41,38,43)(40,47,42,45);(1,3,6,13)(2,7,9,16)(4,11,5,12)(8,15,14,10)(17,19,21,27)(18,22,24,30)(20,26,28,32)(23,29,25,31)(33,35,39,34)(36,42,46,40)(37,43,38,41)(44,47,48,45);(1,4,6,5)(2,8,9,14)(3,11,13,12)(7,15,16,10)(18,23)(20,28)(22,29)(24,25)(26,32)(30,31)(33,36)(34,40)(35,42)(37,45)(38,47)(39,46)(41,48)(43,44);(1,4,6,5)(2,8,9,14)(3,12,13,11)(7,10,16,15)(18,24)(22,30)(23,25)(29,31)(33,37,39,38)(34,41,35,43)(36,45,46,47)(40,48,42,44)

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=g,d^2=gh,e^2=g,f^2=i,b^a=be,c^a=cf,d^a=dhi,e^a=eg,f^a=fi,d^b=dg,e^b=eg,f^b=fh,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, 83695);