#384327

order	512
nilpotency class	3
isoclinism	isoclinic class of #384326
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#14377
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

997 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)(3,14,17,27)(4,18)(5,9,16,11)(6,12)(7,23)(8,13)(10,21)(15,25,30,24)(19,28,29,31)(20,26)(22,32);(1,3)(2,9)(4,15)(5,6)(7,16)(8,17)(10,25)(11,13)(12,27)(14,23)(18,28)(19,20)(21,29)(22,30)(24,26)(31,32);(1,4,7,21,8,22,6,20)(2,10,12,18,13,26,23,32)(3,15,5,19,17,30,16,29)(9,24,14,28,11,25,27,31);(1,5,7,17,8,16,6,3)(2,11,23,14,13,9,12,27)(4,19,21,30,22,29,20,15)(10,25,32,28,26,24,18,31)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 291523);