#180698

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	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#53342	C₂
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

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

normal subgroups only

This group has 691 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) = 256

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

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

Polycyclic presentation:

2^9|a^2=ghi,c^2=h,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, 87894);