#107976

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

754 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=hi,e^2=i,f^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, 15172);