#395986

order	512
nilpotency class	3
isoclinism	isoclinic class of #395986
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#14729
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1319 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)(3,13)(4,7)(5,19,21,18)(6,11)(8,12)(9,15)(10,20,26,22)(14,23,30,32)(16,25)(17,24)(27,28,31,29);(1,3,6,15,8,17,7,16)(2,9,4,13,12,25,11,24)(5,14,20,28,21,30,22,29)(10,23,18,27,26,32,19,31);(1,4,6,12,8,11,7,2)(3,9,15,13,17,25,16,24)(5,18,20,26,21,19,22,10)(14,23,28,27,30,32,29,31);(1,5)(2,10)(3,14)(4,18)(6,20)(7,22)(8,21)(9,23)(11,19)(12,26)(13,27)(15,28)(16,29)(17,30)(24,31)(25,32)

Polycyclic presentation:

2^9|a^2=ef,b^2=e,c^2=e,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,g^b=gi,g^c=gi,h^c=hi,f^d=fi,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, 303182);