#178054

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, 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

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

normal subgroups only

This group has 656 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:

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

Polycyclic presentation:

2^9|a^2=g,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, 85250);