#329601

order	512
nilpotency class	3
isoclinism	isoclinic class of #329587
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 736 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) = 256

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=i,d^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,c^b=cg,g^b=gi,d^c=de,f^c=fh,g^c=gi

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, 236797);