#201501

order	512
nilpotency class	3
isoclinism	isoclinic class of #201424
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

1753 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)(3,12,16,23,13,27,14,8)(7,9,19,24,20,25,18,32)(15,28,31,22,29,26,30,21);(1,3)(2,8,11,23)(4,14)(5,16)(6,13)(7,15)(9,21,25,22)(10,12,17,27)(18,30)(19,31)(20,29)(24,28,32,26);(2,9)(4,18)(5,19)(8,21)(11,25)(14,30)(16,31)(22,23);(2,9)(3,13)(8,22)(10,24)(11,25)(12,26)(14,16)(15,29)(17,32)(21,23)(27,28)(30,31)

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,d^b=dh,f^b=fi,e^c=ei,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72282993, #72283074, #72283617
maximal normal subgroup of	#72282993, #72283074, #72283617
maximal quotient of	#72284190

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 108697);