#463938

order	512
nilpotency class	3
isoclinism	isoclinic class of #463937
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#16497
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

814 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 371134);