#489921

order	512
nilpotency class	4
isoclinism	isoclinic class of #489739
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	#3169	C₂²
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

651 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 651 conjugacy classes of subgroups — too many to draw. Only its 136 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)(3,12)(4,15)(5,11)(6,10)(7,9)(8,13)(14,16)(18,24)(20,22)(21,33)(25,40)(26,44)(27,39)(28,31)(29,45)(32,36)(34,37)(35,47)(38,42)(41,46)(43,48);(1,3,7,4)(2,8,9,16)(5,14,6,13)(10,15,11,12)(17,18)(19,25)(20,26)(21,34)(22,37)(23,27)(24,33)(28,41)(29,46)(30,42)(31,43)(32,39)(35,48)(36,44)(38,47)(40,45);(1,4,7,3)(2,8,9,16)(5,13,6,14)(10,15,11,12)(17,19,23,30)(18,25,27,42)(20,31,22,28)(21,35,36,29)(24,38,39,40)(26,43,37,41)(32,45,33,47)(34,48,44,46);(2,9)(8,16)(10,11)(12,15)(17,20,23,22)(18,26,27,37)(19,28,30,31)(21,32,36,33)(24,34,39,44)(25,41,42,43)(29,45,35,47)(38,46,40,48)

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 397117);