#107537

order	512
nilpotency class	3
isoclinism	isoclinic class of #8510
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#379
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	C₂²⋊#95
prime radical	#3444	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

702 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)(3,12,15,8)(4,16)(5,19)(6,11)(7,10)(9,18)(13,28,29,32)(14,23,20,24)(17,25)(21,26)(22,30,27,31)(33,34,38,42,39,43,37,53)(35,44,46,40,48,55,47,56)(36,49,51,41,52,57,50,58)(45,60,61,59,62,54,63,64);(1,3,6,14,7,15,5,20)(2,8,11,24,10,12,19,23)(4,13,18,31,17,29,21,30)(9,22,25,32,26,27,16,28)(33,35)(34,40)(36,45)(37,47)(38,46)(39,48)(41,54)(42,55)(43,56)(44,53)(49,59)(50,63)(51,61)(52,62)(57,64)(58,60);(1,4)(2,9)(3,13)(5,21)(6,18)(7,17)(8,22)(10,26)(11,25)(12,27)(14,31)(15,29)(16,19)(20,30)(23,28)(24,32)(33,36)(34,41)(35,45)(37,50)(38,51)(39,52)(40,54)(42,57)(43,58)(44,59)(46,61)(47,63)(48,62)(49,53)(55,64)(56,60)

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=i,e^2=h,f^2=i,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=eh,d^b=di,e^b=ehi,f^b=fhi,d^c=dhi,e^c=eh

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 14733);