#481433

order	512
nilpotency class	4
isoclinism	isoclinic class of #481415
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#102
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	C₂×#41	SD₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ C₈×#41 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

705 conjugacy classes of subgroups; only the 240 normal subgroups are shown.

normal subgroups only

This group has 705 conjugacy classes of subgroups — too many to draw. Only its 240 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 388629);