#121856

order	512
nilpotency class	3
isoclinism	isoclinic class of #113366
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#390
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1445
prime radical	C₂⁴	#126
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

395 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)(3,10)(4,7)(5,9)(6,12)(8,14)(11,15)(13,16)(17,18)(19,27,30,23)(20,31)(21,35)(22,25)(24,34)(26,32)(28,43,45,48)(29,38,44,42)(33,41)(36,40)(37,46,39,47);(1,3,5,13)(2,6,9,15)(4,11,14,12)(7,10,8,16)(17,19,21,29,22,30,32,44)(18,23,35,42,25,27,26,38)(20,28,36,47,33,45,34,46)(24,39,31,43,40,37,41,48);(2,7)(3,11)(4,14)(6,15)(8,9)(12,13)(17,20,32,34,22,33,21,36)(18,24,26,41,25,40,35,31)(19,28,44,46,30,45,29,47)(23,37,38,43,27,39,42,48)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 29052);