#6490794

order	512
nilpotency class	3
isoclinism	isoclinic class of #6469989
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#2856	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1578 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)(3,8)(4,7)(5,6)(9,10)(11,14)(12,15)(13,16)(17,18,23,31,22,24,21,25)(19,26,30,20,27,32,28,29);(2,6)(7,8)(9,11,14,10)(12,15,13,16)(17,19,22,27)(18,20,24,29)(21,30,23,28)(25,32,31,26);(1,3,5,4)(2,7,6,8)(10,11)(15,16)(17,20)(18,19)(21,26)(22,29)(23,32)(24,27)(25,28)(30,31);(9,12,14,13)(10,15,11,16)(18,24)(19,27)(25,31)(28,30);(1,4,5,3)(2,7,6,8)(9,13,14,12)(10,15,11,16)(18,24)(20,29)(25,31)(26,32)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6397990);