#165472

order	512
nilpotency class	3
isoclinism	isoclinic class of #165149
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	C₂²×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄²×Q₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

694 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=gh,d^2=ghi,e^2=h,f^2=i,b^a=be,c^a=ch,d^a=dg,e^a=eg,e^b=eh,f^b=fgh,d^c=dgi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 72668);