#6344112

order	512
nilpotency class	3
isoclinism	isoclinic class of #6343406
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	#62815	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2847 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=h,d^2=i,e^2=h,f^2=i,g^2=hi,b^a=bf,c^a=ci,f^a=fh,f^b=fi,g^b=ghi,e^d=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, 6251308);