#482712

order	512
nilpotency class	4
isoclinism	isoclinic class of #482711
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#17595
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂×M₁₆	C₂⁴
prime radical	#60714	C₂
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×#131 ◁ G
LECS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

726 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)(3,13,14,24)(4,17,19,7)(5,8)(6,12)(9,16,26,28)(10,18,22,31)(11,21)(15,27)(20,25)(23,32)(29,30);(2,7)(4,18)(5,21)(8,22)(9,23)(10,11)(12,17)(13,25)(15,29)(16,28)(19,31)(20,32)(24,27)(26,30);(2,8,12,11)(3,14)(4,19)(7,22,17,10)(9,24,26,13)(16,28)(18,31)(23,27,30,25);(1,3,6,14)(2,9)(4,20,19,15)(5,16,21,28)(7,23)(8,13)(10,27)(11,24)(12,26)(17,30)(18,32,31,29)(22,25)

Polycyclic presentation:

2^9|a^2=f,c^2=ghi,d^2=gh,e^2=hi,g^2=i,h^2=i,b^a=be,c^a=ch,d^a=dg,e^a=eg,h^a=hi,e^b=eh,f^b=fghi,g^b=gi,h^b=hi,d^c=di,e^c=ei,e^d=ei,f^d=fi,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, 389908);