#121959

order	512
nilpotency class	3
isoclinism	isoclinic class of #121958
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1445
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#117
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

393 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=de,c^2=de,e^2=hi,g^2=h,b^a=bd,c^a=ce,d^a=dh,f^a=fhi,c^b=cf,d^b=dh,e^b=eh,g^b=gh,d^c=di,e^c=ei,g^c=ghi

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, 29155);