#119427

order	512
nilpotency class	3
isoclinism	isoclinic class of #119426
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1440
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

524 conjugacy classes of subgroups.

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=d,d^2=i,e^2=hi,f^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fhi,c^b=cf,d^b=di,e^b=ehi,f^b=fi,g^b=ghi,e^c=ei,g^c=ghi

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, 26623);