#101229

order	512
nilpotency class	3
isoclinism	isoclinic class of #100939
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#386
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

486 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 486 conjugacy classes of subgroups — too many to draw. Only its 86 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=df,c^2=h,d^2=gh,f^2=ghi,b^a=bd,c^a=ce,d^a=dg,e^a=ei,c^b=cg,d^b=dh,e^b=ei,f^b=fh,d^c=di,f^c=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8425);