#6614749

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574015
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	#37699	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1568 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)(17,18,23,21)(19,27,30,32)(20,26,22,25)(24,31,29,28);(2,6)(7,8)(9,11)(10,13)(12,16)(14,15)(17,19,22,28,23,30,20,31)(18,24,26,27,21,29,25,32);(1,3,5,4)(2,7,6,8)(11,16)(13,14)(17,20,23,22)(18,25,21,26)(19,28,30,31)(24,27,29,32);(1,4,5,3)(2,7,6,8)(9,11)(10,14)(12,16)(13,15)(17,21,23,18)(19,29,30,24)(20,26,22,25)(27,28,32,31);(11,16)(13,14)(17,20,23,22)(18,26,21,25)(19,28,30,31)(24,32,29,27)

Polycyclic presentation:

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

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, 6521945);