#6435912

order	512
nilpotency class	3
isoclinism	isoclinic class of #6433040
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#36644	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2074 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=fg,b^2=g,c^2=f,d^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fi,g^a=gi,c^b=ch,e^b=ei,e^c=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, 6343108);