#6343582

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	#62822	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

2909 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)(3,7)(4,6)(5,8)(9,12)(10,14)(11,13)(15,16)(18,23,25,29)(24,30,32,31);(2,6)(4,12)(7,13)(8,9)(11,16)(14,15)(17,18)(19,24)(20,23)(21,29)(22,25)(26,31)(27,30)(28,32);(1,3)(2,7)(4,11)(5,10)(6,13)(8,14)(9,15)(12,16)(17,19)(18,24)(20,27)(21,26)(22,28)(23,30)(25,32)(29,31);(2,8)(3,10)(6,9)(11,16)(17,20,22,21)(18,23,25,29)(19,26,28,27)(24,31,32,30);(3,10)(7,14)(11,16)(13,15)(17,20,22,21)(18,23,25,29)(19,27,28,26)(24,30,32,31)

Polycyclic presentation:

2^9|a^2=g,d^2=hi,e^2=hi,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.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6250778);