#361341

order	512
nilpotency class	3
isoclinism	isoclinic class of #361285
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2290
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#13895
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×#117 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

877 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=ef,d^2=f,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,g^b=gi,h^b=hi,g^c=gi,h^c=hi,h^d=hi

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, 268537);