#484541

order	512
nilpotency class	4
isoclinism	isoclinic class of #484194
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	C₂²×C₈	C₂⁴
prime radical	C₂⁵	D₈
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×C₂²⋊C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

862 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 862 conjugacy classes of subgroups — too many to draw. Only its 144 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) = 64

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

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

Polycyclic presentation:

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

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, 391737);