#141388

order	512
nilpotency class	4
isoclinism	isoclinic class of #141343
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1957
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#3147	C₂²
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈² ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

324 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 98 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) = 256

A minimal faithful permutation representation — 3 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,c^2=ei,d^2=ghi,e^2=h,f^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,d^b=dh,f^b=fh

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 48584);