#480314

order	512
nilpotency class	4
isoclinism	isoclinic class of #480314
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂×#410
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×M₁₆	C₂⁴
socle	C₂	C₂×#1464
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×C₄² ◁ C₂×C₄×M₁₆ ◁ G
LECS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

725 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 387510);