#489827

order	512
nilpotency class	4
isoclinism	isoclinic class of #489701
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	C₂³	#447
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈×Q₈ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 700 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) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 397023);