#140055

order	512
nilpotency class	4
isoclinism	isoclinic class of #139972
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1889
commutator	C₂×C₂²⋊C₄	C₂²×C₄
Frattini, cosocle	C₂×#117	C₂³
prime radical	#534	C₂×C₄
LCS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#117 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

414 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 414 conjugacy classes of subgroups — too many to draw. Only its 62 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=df,c^2=g,d^2=gi,e^2=g,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=egh,h^a=hi,c^b=ci,d^b=di,e^b=ehi,f^b=fi,d^c=dh,e^c=eg,f^c=fh,e^d=ei,f^e=fi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 47251);