#483932

order	512
nilpotency class	4
isoclinism	isoclinic class of #483928
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#442
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	C₂²×D₈	C₂×C₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

413 conjugacy classes of subgroups; only the 156 normal subgroups are shown.

normal subgroups only

This group has 413 conjugacy classes of subgroups — too many to draw. Only its 156 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=fg,e^2=g,f^2=h,g^2=i,b^a=be,c^a=cg,d^a=dhi,e^a=eg,g^a=gi,c^b=ch,e^b=egi,f^b=fi,g^b=gi,d^c=di,e^c=ei

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, 391128);