#139501

order	512
nilpotency class	4
isoclinism	isoclinic class of #139440
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1898
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×#117	C₂³
prime radical	C₂⁴	#123
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#544 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 405 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=df,c^2=ghi,d^2=gh,e^2=i,f^2=gh,b^a=bd,c^a=ce,d^a=dg,e^a=eghi,g^a=gi,h^a=hi,d^b=dh,e^b=egh,f^b=fh,g^b=gi,h^b=hi,d^c=dgh,e^c=ei,f^c=fgh,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, 46697);