#490741

order	512
nilpotency class	4
isoclinism	isoclinic class of #490699
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#459
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	C₂²×D₈	D₄
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

451 conjugacy classes of subgroups; only the 140 normal subgroups are shown.

normal subgroups only

This group has 451 conjugacy classes of subgroups — too many to draw. Only its 140 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,e^2=g,f^2=hi,h^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fh,h^a=hi,c^b=cg,d^b=di,f^c=fh,h^c=hi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 397937);