#165430

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

703 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)(3,9,11,16)(4,8)(5,7)(6,10,14,12)(13,15)(18,23)(19,26)(20,21)(24,31)(25,29)(30,32)(33,34,37,40,38,41,36,45)(35,42,43,47,44,39,46,48);(1,3)(2,6)(4,12)(5,11)(7,14)(8,16)(9,15)(10,13)(17,18)(19,24)(20,23)(21,29)(22,25)(26,32)(27,31)(28,30)(33,35)(34,39)(36,43)(37,46)(38,44)(40,47)(41,42)(45,48);(2,7)(6,14)(8,15)(9,16)(17,19,22,26)(18,24,25,32)(20,28,21,27)(23,30,29,31);(17,20,22,21)(18,23,25,29)(19,27,26,28)(24,31,32,30)(33,36,38,37)(34,40,41,45)(35,43,44,46)(39,47,42,48)

Polycyclic presentation:

2^9|a^2=f,c^2=gi,d^2=g,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, 72626);