#119918

order	512
nilpotency class	3
isoclinism	isoclinic class of #119915
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1440
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

622 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 622 conjugacy classes of subgroups — too many to draw. Only its 118 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=d,e^2=i,g^2=h,b^a=bd,c^a=ce,d^a=dh,f^a=fi,c^b=cf,e^b=ei,g^b=gh,e^c=ehi,g^c=gi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 27114);