#201492

order	512
nilpotency class	3
isoclinism	isoclinic class of #201414
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	#35050	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1761 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:

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

Polycyclic presentation:

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

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, 108688);