#6495172

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470374
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	#34624	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1649 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)(3,7,8,6)(11,15)(14,16)(17,18)(19,21)(20,28)(22,25)(23,26)(24,31)(27,30)(29,32);(2,5)(6,7)(9,10)(11,14)(12,13)(15,16)(17,19,21,18,23,25,22,26)(20,29,32,28,31,27,30,24);(1,3,4,8)(2,6,5,7)(10,13)(14,16)(17,20,23,31)(18,24,26,28)(19,27,25,29)(21,32,22,30);(10,13)(14,16)(17,21,23,22)(18,19,26,25)(20,30,31,32)(24,29,28,27);(9,11,12,15)(10,14,13,16)(17,21,23,22)(18,25,26,19)(20,30,31,32)(24,27,28,29)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6402368);