#176330

order	512
nilpotency class	3
isoclinism	isoclinic class of #176173
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

643 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 643 conjugacy classes of subgroups — too many to draw. Only its 142 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 83526);