#6344654

order	512
nilpotency class	3
isoclinism	isoclinic class of #60372
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₄²	C₂⁵
socle	C₂²	C₂²×C₄×D₄
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#431 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1696 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

122 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6251850);