#6344217

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	C₄²	C₂⁵
prime radical	#62838	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ #3444 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2681 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,d^2=h,f^2=i,g^2=hi,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.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6251413);