#121252

order	512
nilpotency class	3
isoclinism	isoclinic class of #121250
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1443
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#126
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

370 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:

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

Polycyclic presentation:

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

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, 28448);