#6410533

order	512
nilpotency class	2
isoclinism	isoclinic class of #37644
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×#121	C₂×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1528 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=f,d^2=g,e^2=g,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cgi,d^b=df

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, 6317729);