#6344228

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	#62836	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ #3444 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2863 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:

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

Polycyclic presentation:

2^9|a^2=g,d^2=hi,e^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, 6251424);