#6343579

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	#62828	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

2782 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:

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

Polycyclic presentation:

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6250775);