#6343525

order	512
nilpotency class	3
isoclinism	isoclinic class of #6343524
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	#62819	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

3078 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6250721);