#180756

order	512
nilpotency class	3
isoclinism	isoclinic class of #180581
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

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

normal subgroups only

This group has 574 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) = 64
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 64 points:

(2,6)(4,11)(7,15)(8,12)(10,13)(14,16)(19,26)(20,21)(23,30)(24,28)(27,29)(31,32)(33,34,39,44)(35,45,48,58)(36,41,50,56)(37,43,38,42)(40,52,54,47)(46,59,62,63)(49,57,51,55)(53,60,64,61);(1,2,5,8)(3,7,9,16)(4,12,11,6)(10,14,13,15)(17,18)(19,23)(20,24)(21,28)(22,25)(26,31)(27,32)(29,30)(33,35,39,48)(34,40,44,54)(36,46,50,62)(37,47,38,52)(41,53,56,64)(42,45,43,58)(49,60,51,61)(55,63,57,59);(1,3,5,9)(2,7,8,16)(4,13,11,10)(6,14,12,15)(17,19)(18,23)(20,26)(21,29)(22,27)(24,31)(25,32)(28,30)(33,36)(34,41)(35,46)(37,51)(38,49)(39,50)(40,53)(42,57)(43,55)(44,56)(45,59)(47,61)(48,62)(52,60)(54,64)(58,63);(2,8)(3,9)(6,12)(10,13)(17,20,22,21)(18,24,25,28)(19,26,27,29)(23,31,32,30)(33,37,39,38)(34,42,44,43)(35,47,48,52)(36,49,50,51)(40,45,54,58)(41,55,56,57)(46,60,62,61)(53,63,64,59)

Polycyclic presentation:

2^9|a^2=ghi,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, 87952);