#483223

order	512
nilpotency class	4
isoclinism	isoclinic class of #482075
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#102
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	C₂×#3169	C₂
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

880 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 32 points:

(1,2,5,10,6,8,4,9)(3,11,12,16,13,7,14,15)(17,18,20,23,22,26,19,27,21,24,28,31,29,32,30,25);(1,3)(2,7)(4,12)(5,14)(6,13)(8,11)(9,15)(10,16)(18,23)(19,28)(20,30)(22,29)(24,31)(25,26)(27,32);(2,8)(7,11)(9,10)(15,16);(1,4,6,5)(2,9,8,10)(3,12,13,14)(7,15,11,16)(18,24)(23,31)(25,27)(26,32)

Polycyclic presentation:

2^9|a^2=f,d^2=gi,e^2=hi,f^2=ghi,h^2=i,b^a=be,c^a=cgi,d^a=di,e^a=eg,e^b=eh,f^b=fgh,h^b=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285321, #72285335, #72285367, …
maximal normal subgroup of	#72285321, #72285335, #72285367, …
maximal quotient of	#72286285

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 390419);