#164124

order	512
nilpotency class	3
isoclinism	isoclinic class of #163983
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

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	C₂×#121	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

725 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)(3,9,11,6)(4,7)(5,8)(10,15,16,13)(12,14)(17,18)(19,24)(20,25)(21,30)(22,26)(23,27)(28,31)(29,32)(33,34,37,41,38,42,36,40)(35,39,45,48,43,46,44,47);(1,3,5,11)(2,6,8,9)(4,10,12,16)(7,13,14,15)(17,19)(18,23)(20,27)(21,29)(22,28)(24,30)(25,31)(26,32)(33,35)(34,39)(36,44)(37,45)(38,43)(40,47)(41,48)(42,46);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(17,19)(18,24)(20,29)(21,27)(22,28)(23,30)(25,32)(26,31)(35,43)(39,46)(44,45)(47,48);(17,20,22,21)(18,25,26,30)(19,27,28,29)(23,31,32,24)(33,36,38,37)(34,40,42,41)(35,44,43,45)(39,47,46,48)

Polycyclic presentation:

2^9|a^2=f,b^2=gh,c^2=gh,d^2=hi,e^2=h,f^2=i,b^a=be,e^a=eg,c^b=ci,e^b=eh,f^b=fgh,d^c=dh

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 71320);