#461300

order	512
nilpotency class	3
isoclinism	isoclinic class of #461299
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#16386
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

856 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:

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

Polycyclic presentation:

2^9|a^2=efg,b^2=ef,d^2=ei,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,g^b=gi,d^c=de,e^c=ei,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, 368496);