#107981

order	512
nilpotency class	3
isoclinism	isoclinic class of #107966
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

740 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=h,d^2=hi,e^2=i,f^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=gh,d^b=di,e^b=eh,f^b=fi,d^c=dh,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, 15177);