#178053

order	512
nilpotency class	3
isoclinism	isoclinic class of #177893
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	C₂×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

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

normal subgroups only

This group has 730 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) = 256

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

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

Polycyclic presentation:

2^9|b^2=gh,c^2=i,d^2=h,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,d^c=dg,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, 85249);