#197584

order	512
nilpotency class	3
isoclinism	isoclinic class of #19186
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#386
prime radical	#60900	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1139 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72215776, #72215973, #72216030
maximal normal subgroup of	#72215776, #72215973, #72216030
maximal quotient of	#72215714, #72215720

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 104780);