#481729

order	512
nilpotency class	4
isoclinism	isoclinic class of #481729
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂×#1505
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂×M₁₆	C₂⁴
prime radical	#60710	C₂
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #2873 ◁ G
LECS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

774 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)(3,13,14,26)(4,17,21,7)(5,10)(6,12)(8,16,24,28)(9,18,23,32)(11,20)(15,27)(19,25)(22,31)(29,30);(2,7)(4,18)(5,20)(8,22)(9,11)(10,23)(12,17)(13,25)(15,29)(16,28)(19,31)(21,32)(24,30)(26,27);(1,3,6,14)(2,8)(4,19,21,15)(5,16,20,28)(7,22)(9,27)(10,13)(11,26)(12,24)(17,30)(18,31,32,29)(23,25);(3,14)(8,24)(13,26)(15,19)(16,28)(22,30)(25,27)(29,31)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 388925);