#107978

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	#3444	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

697 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)(3,11)(4,15,17,8)(5,18)(6,10)(7,13)(9,20)(12,25,28,32)(14,26)(16,23,19,24)(21,27,31,29)(22,30)(33,34,37,44,39,42,38,43)(35,40,47,55,46,56,52,54)(36,48,51,57,49,58,50,41)(45,59,61,64,62,63,60,53);(1,3)(2,7,10,22)(4,12)(5,13)(6,14)(8,21,15,31)(9,11,18,26)(16,27)(17,28)(19,29)(20,30)(23,32,24,25)(33,35,38,47,39,46,37,52)(34,40,44,54,42,56,43,55)(36,45,50,61,49,62,51,60)(41,53,48,63,57,64,58,59);(1,4)(2,8)(3,12)(5,19)(6,17)(7,21)(9,24)(10,15)(11,25)(13,29)(14,28)(16,20)(18,23)(22,31)(26,32)(27,30)(33,36)(34,41)(35,45)(37,51)(38,50)(39,49)(40,53)(42,57)(43,58)(44,48)(46,62)(47,61)(52,60)(54,63)(55,59)(56,64)

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=hi,e^2=i,f^2=i,g^2=i,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, 15174);