#121669

order	512
nilpotency class	3
isoclinism	isoclinic class of #111436
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#390
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1445
prime radical	C₂⁴	C₄×Q₈
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

416 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 126 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=deh,c^2=de,d^2=i,f^2=i,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fi,c^b=cf,d^b=di,e^b=ei,f^b=fi,g^b=ghi,f^c=fi,g^c=gh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 28865);