#321990

order	512
nilpotency class	3
isoclinism	isoclinic class of #321983
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

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

649 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 649 conjugacy classes of subgroups — too many to draw. Only its 150 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:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=egi,d^2=eh,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=dei,e^a=eh,c^b=cg,d^b=dh,f^c=fi,g^c=gh

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, 229186);