#120804

order	512
nilpotency class	3
isoclinism	isoclinic class of #120804
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1442
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#123
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

408 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 116 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) = 64

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=d,d^2=i,f^2=h,b^a=bd,c^a=ce,d^a=dh,c^b=cf,d^b=di,g^b=ghi,e^c=eh,f^c=fhi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 28000);