#139835

order	512
nilpotency class	4
isoclinism	isoclinic class of #139724
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1898
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×#41	C₂³
prime radical	C₂×#537	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#537 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂³ ▷ C₂ ▷ C₁

544 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 544 conjugacy classes of subgroups — too many to draw. Only its 68 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)(4,11,12,7)(5,8)(9,13,15,16)(10,14)(17,18,22,27)(19,28)(20,34)(21,26,38,43)(23,30,41,31)(24,37)(25,29)(32,45)(33,46)(35,42)(36,39)(40,47,44,48);(1,3,5,10)(2,6,8,14)(4,9,12,15)(7,13,11,16)(17,19,29,31,22,33,20,30)(18,23,34,46,27,41,25,28)(21,36,37,47,38,32,35,48)(24,39,26,44,42,45,43,40);(1,4)(2,7)(3,9)(5,12)(6,13)(8,11)(10,15)(14,16)(17,20,22,29)(18,24,27,42)(19,30,33,31)(21,37,38,35)(23,39,41,45)(25,43,34,26)(28,40,46,44)(32,48,36,47)

Polycyclic presentation:

2^9|a^2=f,b^2=dfh,c^2=i,e^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ehi,g^a=gi,d^b=dg,e^b=eh,f^b=fg,g^b=gi,d^c=dh,e^c=ei,f^c=fh,f^d=fi

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 47031);