#6476550

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470395
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#35301	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1743 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=i,e^2=g,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,e^c=ei,e^d=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6383746);