#6614745

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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1537 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)(17,18,22,25)(19,26,24,21)(20,23,28,30)(27,32,31,29);(2,6)(7,8)(9,11)(10,13)(12,16)(14,15)(17,19,21,18,22,24,26,25)(20,27,29,23,28,31,32,30);(1,3,5,4)(2,7,6,8)(11,16)(13,14)(17,20)(18,23)(19,27)(21,29)(22,28)(24,31)(25,30)(26,32);(1,4,5,3)(2,7,6,8)(9,11)(10,14)(12,16)(13,15)(20,28)(23,30)(27,31)(29,32);(11,16)(13,14)

Polycyclic presentation:

2^9|a^2=i,b^2=fgh,c^2=g,d^2=g,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, 6521941);