#177814

order	512
nilpotency class	3
isoclinism	isoclinic class of #177733
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

670 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 670 conjugacy classes of subgroups — too many to draw. Only its 132 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=g,d^2=gh,e^2=g,f^2=i,b^a=be,c^a=cf,d^a=di,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

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 85010);