#139904

order	512
nilpotency class	4
isoclinism	isoclinic class of #139783
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	C₂²×Q₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×Q₈	C₂³
prime radical	C₂³	#386
LCS	G ▷ C₂²×Q₈ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×Q₈ ◁ G
LECS	G ▷ C₂×C₄×Q₈ ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 397 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,c^2=h,d^2=i,e^2=i,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=ehi,h^a=hi,c^b=ci,d^b=dgi,e^b=ehi,f^b=fgi,d^c=dh,f^c=fh,e^d=ei,f^e=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, 47100);