#6344452

order	512
nilpotency class	3
isoclinism	isoclinic class of #61569
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	C₂³×C₄⋊C₄
prime radical	C₂²×#41	Q₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1566 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,e^2=h,f^2=h,g^2=i,b^a=bf,f^a=fh,c^b=ch,e^b=ehi,d^c=dhi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6251648);