#6343561

order	512
nilpotency class	3
isoclinism	isoclinic class of #6343554
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂³×C₂²⋊C₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#62821	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

3088 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=g,d^2=i,f^2=i,b^a=bf,d^a=dh,f^a=fh,f^b=fi,g^b=ghi,d^c=di,e^c=eh

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal quotient of

#72239068

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6250757);