#6343727

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	#62835	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#532 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

2945 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,e^2=hi,f^2=i,b^a=bf,d^a=dh,e^a=ei,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, 6250923);