#321975

order	512
nilpotency class	3
isoclinism	isoclinic class of #321957
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2541
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#532	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

645 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 645 conjugacy classes of subgroups — too many to draw. Only its 162 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=fhi,b^2=eg,d^2=ehi,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=dei,e^a=eh,c^b=cg,f^c=fi,g^c=gh

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 229171);