#164907

order	512
nilpotency class	3
isoclinism	isoclinic class of #164127
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

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

681 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=gi,d^2=gi,e^2=h,f^2=i,b^a=be,e^a=eg,e^b=eh,f^b=fgh,d^c=dghi

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 72103);