#490350

order	512
nilpotency class	4
isoclinism	isoclinic class of #490115
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	#33783	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

680 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 680 conjugacy classes of subgroups — too many to draw. Only its 134 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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=h,d^2=ghi,e^2=gi,f^2=hi,g^2=i,b^a=be,c^a=cf,d^a=dgh,e^a=eg,f^a=fhi,g^a=gi,d^b=dg,e^b=eg,f^b=fh,g^b=gi,e^c=eh,f^c=fhi

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, 397546);