#119055

order	512
nilpotency class	3
isoclinism	isoclinic class of #10524
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#394
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1440
prime radical	C₂⁴	#126
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

480 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 64 points:

(1,2,6,12,7,10,5,11)(3,13,16,24,17,8,15,25)(4,18,21,28,20,9,22,27)(14,23,19,26,30,29,31,32)(33,34)(35,45)(36,41,52,60)(37,44)(38,42)(39,43)(40,47)(46,57,63,61)(48,55)(49,56)(50,54,62,64)(51,59,53,58);(1,3,7,17)(2,8,10,13)(4,19,20,31)(5,16,6,15)(9,26,18,32)(11,25,12,24)(14,22,30,21)(23,28,29,27)(33,35,39,48)(34,40,43,56)(36,50,52,62)(37,49,38,47)(41,57,60,61)(42,55,44,45)(46,51,63,53)(54,58,64,59);(1,4)(2,9,10,18)(3,14)(5,22)(6,21)(7,20)(8,23,13,29)(11,28,12,27)(15,31)(16,19)(17,30)(24,32,25,26)(33,36,37,51,39,52,38,53)(34,41,42,58,43,60,44,59)(35,46,47,62,48,63,49,50)(40,54,45,61,56,64,55,57)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=d,d^2=i,f^2=h,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fi,c^b=cf,d^b=di,e^b=ei,f^b=fh,g^b=ghi,e^c=eh,f^c=fhi,g^c=gi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 26251);