#6344779

order	512
nilpotency class	3
isoclinism	isoclinic class of #60771
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	#60572	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₈×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1587 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=gh,f^2=h,g^2=i,b^a=bf,c^a=ch,d^a=di,f^a=fh,c^b=ci,f^b=fh,e^c=eh,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, 6251975);