#112150

order	512
nilpotency class	3
isoclinism	isoclinic class of #112147
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1438
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	#60801	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 853 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)(3,9,11,15)(4,7)(5,8)(6,10,14,12)(13,16)(17,18,22,27,23,26,21,36)(19,28,32,44,30,24,31,39)(20,25,34,43,35,41,37,42)(29,38,33,48,46,45,47,40);(1,3)(2,6)(4,12)(5,11)(7,15)(8,14)(9,16)(10,13)(17,19)(18,24)(20,33)(21,32)(22,31)(23,30)(25,40)(26,28)(27,44)(29,34)(35,47)(36,39)(37,46)(38,42)(41,48)(43,45);(4,13)(6,14)(7,16)(10,12)(17,20)(18,25)(19,29)(21,37)(22,34)(23,35)(24,38)(26,41)(27,43)(28,45)(30,46)(31,47)(32,33)(36,42)(39,48)(40,44)

Polycyclic presentation:

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

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, 19346);