#139856

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

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

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

normal subgroups only

This group has 548 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)(17,18,23,28,24,29,22,38)(19,30,33,41,34,47,39,46)(20,27,35,26,21,36,31,44)(25,40,42,37,43,48,45,32);(1,3,5,10)(2,6,8,14)(4,9,12,15)(7,13,11,16)(17,19,31,40)(18,25,26,41)(20,32,22,33)(21,37,23,39)(24,34,35,48)(27,30,28,42)(29,43,44,46)(36,47,38,45);(1,4,5,12)(2,7,8,11)(3,9,10,15)(6,13,14,16)(17,20)(18,26)(19,32)(21,24)(22,31)(23,35)(25,41)(27,38)(28,36)(29,44)(30,45)(33,40)(34,37)(39,48)(42,47)(43,46)

Polycyclic presentation:

2^9|a^2=f,b^2=dfh,c^2=h,e^2=i,f^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, 47052);