#142355

order	512
nilpotency class	4
isoclinism	isoclinic class of #142242
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	C₂²×#41	D₄
LCS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#408 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

450 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 450 conjugacy classes of subgroups — too many to draw. Only its 70 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)(17,18,21,26)(19,28,33,47)(20,34,37,45)(22,27,23,40)(24,41,43,30)(25,38,36,39)(29,46,42,35)(31,44,32,48);(1,3,5,10)(2,6,8,13)(4,9,11,16)(7,14,15,12)(17,19,21,33)(18,24,26,43)(20,35,37,46)(22,32,23,31)(25,44,36,48)(27,29,40,42)(28,45,47,34)(30,38,41,39);(1,4)(2,7)(3,9)(5,11)(6,12)(8,15)(10,16)(13,14)(17,18)(19,29)(20,36)(21,26)(22,40)(23,27)(24,32)(25,37)(28,46)(30,44)(31,43)(33,42)(34,38)(35,47)(39,45)(41,48)

Polycyclic presentation:

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

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 49551);