#182863

order	512
nilpotency class	3
isoclinism	isoclinic class of #17603
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×C₄×C₂²⋊C₄
prime radical	#3444	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×#97 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

951 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=f,d^2=i,e^2=h,f^2=h,b^a=be,e^a=eh,g^a=ghi,d^b=dh,e^b=ei,d^c=dh

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 90059);