#483313

order	512
nilpotency class	4
isoclinism	isoclinic class of #483313
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#17754
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	#60725	C₂
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #2875 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

677 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 390509);