#6380184

order	512
nilpotency class	3
isoclinism	isoclinic class of #61109
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂×D₄²
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂	C₂×D₄×C₄⋊C₄
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2419 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=h,c^2=g,f^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,d^b=df,e^b=ei,f^b=fi,e^c=ei,f^d=fi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6287380);