#179427

order	512
nilpotency class	3
isoclinism	isoclinic class of #179229
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

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

normal subgroups only

This group has 660 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)(3,10)(4,7)(5,9)(6,15)(8,13)(11,12)(14,16)(22,28)(29,32)(35,41)(36,43)(38,46)(40,44)(42,45)(47,48);(1,3,7,14)(2,8,4,11)(5,13,6,12)(9,16,15,10)(17,18,21,24)(19,22,26,28)(20,23,27,30)(25,29,31,32)(33,34)(35,38)(36,44)(37,39)(40,43)(41,48)(42,47)(45,46);(1,4,7,2)(3,11,14,8)(5,9,6,15)(10,13,16,12)(17,19)(18,22)(20,25)(21,26)(23,29)(24,28)(27,31)(30,32)(33,35)(34,38)(36,45)(37,42)(39,47)(40,48)(41,43)(44,46);(1,5,7,6)(2,9,4,15)(3,12,14,13)(8,10,11,16)(17,20,21,27)(18,23,24,30)(19,25,26,31)(22,29,28,32)(34,39)(38,47)(40,44)(46,48)

Polycyclic presentation:

2^9|b^2=ghi,c^2=g,d^2=ghi,e^2=g,f^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fi,d^b=dgi,e^b=eg,f^b=fh,d^c=dg,e^c=eh,f^c=fi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 86623);