#119919

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

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

normal subgroups only

This group has 606 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,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, 27115);