#481872

order	512
nilpotency class	4
isoclinism	isoclinic class of #481819
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#99
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄⋊C₄	C₂⁴
prime radical	C₂⁵	C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

969 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=i,e^2=hi,f^2=i,b^a=be,c^a=ch,e^a=eg,g^a=gi,d^b=di,e^b=eh,f^b=fghi,f^e=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 389068);