#112848

order	512
nilpotency class	3
isoclinism	isoclinic class of #112845
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	C₂×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

848 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 848 conjugacy classes of subgroups — too many to draw. Only its 138 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:

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

Polycyclic presentation:

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

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, 20044);