#108313

order	512
nilpotency class	3
isoclinism	isoclinic class of #8465
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	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	C₂²⋊#95
prime radical	#3444	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

718 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:

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

Polycyclic presentation:

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

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, 15509);