#200411

order	512
nilpotency class	3
isoclinism	isoclinic class of #200406
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#35051	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1399 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)(3,15,20,30)(4,17,23,32)(5,12)(6,27)(7,28)(8,13)(9,14)(10,18,24,11)(16,25)(19,21,29,22)(26,31);(1,3,6,11,7,29,26,32)(2,10,12,30,28,4,16,21)(5,17,9,20,25,18,8,19)(13,23,27,22,14,24,31,15);(1,4)(2,11)(3,16)(5,15)(6,21)(7,10)(8,24)(9,23)(12,29)(13,17)(14,18)(19,31)(20,27)(22,25)(26,30)(28,32);(1,5,7,25)(2,12,28,16)(3,17,29,18)(4,21,10,30)(6,9,26,8)(11,20,32,19)(13,27,14,31)(15,23,22,24)

Polycyclic presentation:

2^9|a^2=g,b^2=eg,d^2=hi,e^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fi,e^b=ei,f^b=fi,g^b=gi,d^c=dh,e^c=ei,g^c=gi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 107607);