#444866

order	512
nilpotency class	3
isoclinism	isoclinic class of #421157
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2610
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15791
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

618 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 352062);