#120312

order	512
nilpotency class	3
isoclinism	isoclinic class of #120311
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1441
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#124
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

448 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 448 conjugacy classes of subgroups — too many to draw. Only its 130 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)(3,10)(4,8)(5,7)(6,11)(9,13)(12,16)(14,15)(17,18)(19,29)(20,33,36,43)(21,27)(22,26)(23,28)(24,31)(25,35,44,38)(30,39,46,48)(32,40)(34,42,45,47)(37,41);(1,3,4,11)(2,6,8,10)(5,14,13,12)(7,16,9,15)(17,19,21,31,23,32,22,37)(18,24,26,29,28,41,27,40)(20,34,35,30,36,45,38,46)(25,42,43,48,44,47,33,39);(2,7)(5,13)(6,15)(8,9)(10,16)(12,14)(17,20,21,35,23,36,22,38)(18,25,26,43,28,44,27,33)(19,30,31,45,32,46,37,34)(24,39,29,42,41,48,40,47)

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=di,d^2=h,e^2=hi,f^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fi,c^b=cf,e^b=ei,f^b=fh,e^c=ei,f^c=fh,g^c=gh

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, 27508);