#6408197

order	512
nilpotency class	2
isoclinism	isoclinic class of #42996
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₄	C₂⁵
commutator, socle	C₂³	C₂⁴×C₄
prime radical	C₂×#392	C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2041 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)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18,22,26,21,23,20,25)(19,24,27,31,28,30,29,32);(2,5)(3,7)(9,11)(10,13)(12,15)(14,16)(18,23)(24,30)(25,26)(31,32);(10,14)(11,15)(17,19)(18,24)(20,29)(21,28)(22,27)(23,30)(25,32)(26,31);(1,3,4,7)(2,6,5,8)(10,14)(13,16)(17,20,21,22)(18,25,23,26)(19,27,28,29)(24,31,30,32);(2,5)(6,8)(18,23)(24,30)(25,26)(31,32)

Polycyclic presentation:

2^9|a^2=h,d^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dfi,d^c=di

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6315393);