#107899

order	512
nilpotency class	3
isoclinism	isoclinic class of #107898
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

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	C₂×#121	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

726 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 15095);