#489884

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	#33669	C₂
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

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

normal subgroups only

This group has 711 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)(3,12)(4,15)(5,11)(6,10)(7,9)(8,13)(14,16)(17,18)(19,28)(20,25)(21,33)(22,37)(23,26)(24,30)(27,35)(29,45)(31,46)(32,42)(34,39)(36,44)(38,43)(40,47)(41,48);(1,3,7,4)(2,8,9,16)(5,14,6,13)(10,15,11,12)(17,19,23,31)(18,24,26,40)(20,29,22,38)(21,34,36,48)(25,39,37,41)(27,43,42,45)(28,44,46,33)(30,32,47,35);(1,4,7,3)(2,8,9,16)(5,13,6,14)(10,15,11,12)(17,20,23,22)(18,25,26,37)(19,29,31,38)(21,32,36,35)(24,39,40,41)(27,33,42,44)(28,45,46,43)(30,34,47,48);(2,9)(8,16)(10,11)(12,15)(18,26)(24,40)(25,37)(27,42)(28,46)(33,44)(39,41)(43,45)

Polycyclic presentation:

2^9|b^2=hi,c^2=hi,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.

Relationships

maximal subgroup of	#72285609
maximal normal subgroup of	#72285609

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 397080);