#114855

order	512
nilpotency class	3
isoclinism	isoclinic class of #10004
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#392
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1438
prime radical	#3444	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

792 conjugacy classes of subgroups.

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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 22051);