#108011

order	512
nilpotency class	3
isoclinism	isoclinic class of #108006
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²⋊#95
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#120
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

736 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 15207);