#6963987

order	512
nilpotency class	3
isoclinism	isoclinic class of #6884809
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2085 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|c^2=f,d^2=i,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,d^c=dfh,g^c=gi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6871183);