#164116

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)(19,23)(20,22)(21,25)(24,28)(26,30)(27,29)(31,32)(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)(18,22)(20,28)(23,29)(24,25)(27,32)(30,31)(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)(18,23)(20,27)(21,26)(22,29)(24,31)(25,30)(28,32)(35,43)(39,46)(44,45)(47,48);(19,26)(23,30)(27,32)(29,31)(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=i,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, 71312);