#142370

order	512
nilpotency class	4
isoclinism	isoclinic class of #142252
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1999
commutator, Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂⁴	#135
LCS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#424 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

397 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 397 conjugacy classes of subgroups — too many to draw. Only its 72 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 — 3 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=g,d^2=gi,g^2=i,b^a=bd,c^a=ce,d^a=dg,f^a=fh,g^a=gi,c^b=cf,d^b=dg,g^b=gi,d^c=dh,e^c=ei,f^c=fi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 49566);