#489917

order	512
nilpotency class	4
isoclinism	isoclinic class of #489733
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×#446 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

654 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 654 conjugacy classes of subgroups — too many to draw. Only its 138 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)(17,18,23,27)(19,28,31,46)(20,25,22,37)(21,33,36,44)(24,39,41,30)(26,32,45,35)(29,47,38,43)(34,40,48,42);(1,3,7,4)(2,8,9,16)(5,14,6,13)(10,15,11,12)(17,19)(18,24)(20,29)(21,34)(22,38)(23,31)(25,40)(26,43)(27,41)(28,33)(30,35)(32,39)(36,48)(37,42)(44,46)(45,47);(1,4,7,3)(2,8,9,16)(5,13,6,14)(10,15,11,12)(17,20,23,22)(18,25,27,37)(19,29,31,38)(21,35,36,32)(24,40,41,42)(26,44,45,33)(28,43,46,47)(30,48,39,34);(2,9)(8,16)(10,11)(12,15)

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=h,e^2=gi,f^2=hi,g^2=i,b^a=be,c^a=cf,d^a=dhi,e^a=eg,f^a=fhi,g^a=gi,e^b=eg,f^b=fh,g^b=gi,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, 397113);